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1983 Electronic Structure of Iron. Diola Bagayoko Louisiana State University and Agricultural & Mechanical College
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Bagayoko, Diola
ELECTRONIC STRUCTURE OF IRON
The Louisiana State University and Agricultural and Mechanical Col. Ph.D. 1983
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University Microfilms International ELECTRONIC STRUCTURE OF IRON
A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
in
The Department of Physics and Astronoiny
by
Diola Bagayoko B.S., Ecole Normale Superieure de Bamako, Mali, 1973 M.S., Lehigh University, Bethlehem, Pennsylvania, USA, 1978 August 1983 DEDICATION
As an expression of my gratitude for their love and support, I dedicate this dissertation to my mother, Nagnouma (kind mother)
Keita; my father, Djigui (the reliant) Bagayoko; my grandfather,
Namory (the great leader) Keita; my grandmother, Mariam Traore; my wife, Ella Kelley; and my son, Namory Djigui Bagayoko.
I extend this dedication to the memories of my ancestor,
Professor "Cheik" Mohamed Bagayoko of the University of Timbuctu, whose hard work, intellectual accomplishments, and great human qualities are a continual source of inspiration for me. In doing so
I am hereby presenting this dissertation to my sons and daughters, present and future, as an invitation to uphold the family tradition.
ii ACKNOWLEDGEMENTS
I am deeply grateful to Professor Joseph Callaway for supporting and guiding me throughout this research. I have immensely benefited from his broad and profound understanding of physics. His personal habit of working hard, material and moral assistances constituted a quintessence of the successful completion of this endeavor and serve as inspirational examples for me in my career as a scientist.
I thank the African-American Institute, Lehigh and Brandeis
Universities for the gracious AFGRAD fellowship which enabled me to undertake this Ph.D. program. The financial support of Louisiana
State University and the U.S. National Science Foundation for large computations as well as participation to several conferences i to be commended. Dr. Charles E. Coates Memorial Fund of The L.S.U.
Foundation assisted for the publication of this dissertation.
Faculty members, post-doctoral researchers and students of the
Department of Physics and Astronoiry provided me with a stimulating academic atmosphere I highly appreciated. I particularly thank
Professors A. K. Rajagopal, Jerry Draayer, Claude Grenier, A. R. P.
Rau, Roy Goodrich, Ronald J. Henry, R. G. Hussey, and Drs. Xianwu
Zou, Alfred Ziegler, S. P. Singhal, Uday Gupta, M. V. Ramana, Alfred
Msezane, Samir Shattacharya, Lou Adams, J. Perez-Mercader, Eugene Ho, and Graduate Students Khachig Jerjian, S. Dhar, Gonzalo Fuster, Dipak
Oza, and Yosua Namba for their help and support. I am indebted to
Professor Norman M. Mach for invaluable suggestions. I thank
Professor Alan Marshak for sparing some of his time for me. I am
i ii grateful to Dr. D. G. Laurent who taught me the essentials of the
original band package and to Dr. Nathalie Zongo N'Guessan for her
support. The L.S.U. System Network Computer Center and the System
Analyst Ms. Hortensia Delgado played a key role in the success of r\y
interface with the IBM 3033. To help, and always with pleasure is a
distinction Ms. Delgado has earned. Artist N. P. Harris kindly
reinked and photographed the figures in this document. The typing
services of Linda Gauthier and Daisy Mehrotra have been very helpful
in the production of this publication.
I finally wish to express rry profound gratitude to Dj-gui
Bagayoko, Namory Keita, Ella Kelley, C. M. Cherif Keita, and families for their love and encouragement. TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ...... iii
LIST OF TABLES...... viii
LIST OF FIGURES...... ix
ABSTRACT...... x
CHAPTER
1 INTRODUCTION ...... 1
2 GENERAL THEORY ...... 8
A. The LCAO Method ...... 8
B. The RSK Local Density Functional
P o te n tia l ...... 11
C. Evaluation of the Total Energy of a
M etal...... 14
3 COMPUTATIONAL PROCEDURE ...... 18
A. BNDPKG...... 18
B. Contractions ...... 21
4 ELECTRONIC STRUCTURES OFBCC AND FCC IRON...... 31
A. BCC Iron, a = 5.4057 a.u ...... 31
B. Volume Dependence of BCC Iron Band
S tru ctu re...... 36
C. Electronic Structure of FCC I r o n ...... 42
D. BCC and FCC Iron Form F ac to rs ...... 49
5 MAGNETISM IN IRON...... 58
A. Ferromagnetism in BCC and FCC I r o n ...... 58
B. Comparison with Experim ent ...... 68
v C. Relevanceto Iron Based Al l o y s ...... 70
6 DISCUSSION AND SUMMARY...... 73
A. Discussion forFuture W o rk ...... 73
B. Summary...... 76
REFERENCES...... 79
APPENDIX A ...... 85
Appendix A.l: New version of PROGRAM FC O F ...... 86
Appendix A.2: New version of PROGRAM S C F 2 ...... 113
Appendix A.3: New version of PROGRAM DENST ..... 126
APPENDIX B ...... 138
Appendix B.l: Energy levels for BCC iron at high
symmetry points, a = 5.2 a.u. . . . 139
Appendix B.2: Energy levels for BCC iron at high
symmetry points, a = 5.4057 a.u. . . 140
Appendix B.3: Energy levels for BCC iron at high
symmetry points, a = 5.6 a.u. . . . 141
Appendix B.4: Energy levels for FCC iron at high
symmetry points, a = 6.5516 a.u. . . 142
Appendix B.5: Energy levels for FCC iron at high
symmetry points, a = 6.8107 a.u. . . 143
Appendix B.6: Energy levels for FCC iron at high
symmetry points, a : 7,0 a.u. . . . 144
Appendix B.7: Least square fit coefficients
for BCC iron charge form factors
around a =5.4057 a.u ...... 145
vi Appendix B.8: Least square f i t coefficients
for BCC iron spin form factors
around a =5.4057 a . u ...... 146
APPENDIX C ...... 147
Appendix C.l: Evaluation of the pressure for a metal. 148
VITA ...... 150
vii LIST OF TABLES
Table Page
I. Contractions of Gaussian basis s e t ...... 25
II. Comparison of contractions ...... 26
III. Representative band widths and exchange splittings for BCC iron at various atomic volumes ...... 34
IV. Representative band widths and exchange splittings for BCC iron at various atomic volumes; a supplement ...... 35
V. Representative band widths and exchange splittings for FCC iron at various atomic volumes ...... 47
VI. Representative band widths and exchange splittings for FCC iron at various atomic volumes; a supplement ...... 48
VII. Charge and spin form factors for BCC iron at a lattice constant of 5.0 a.u ...... 51
VIII. Charge and spin form factors for BCC iron at a lattice constant of 5.4057 a.u...... 52
IX. Charge and spin form factors for BCC iron at a lattice constant of 6.0 a.u ...... 53
X, Charge and spin form factors for FCC iron at a lattice constant of 6.5516 a.u ...... 54
XI. Charge and spin form factors for FCC iron at a lattice constant of 6.8107 a.u ...... 55
XII. Charge and spin form factors for FCC iron at a lattice constant of 7.0 a.u ...... 56
XIII. Magnetic moment, characteristic splittings and density of states at the Fermi level for BCC and FCC iron at various atomic volumes ...... 60
XIV. Magnetic moment of BCC cobalt at various atomic volumes ...... 65
viii LIST OF FIGURES
Figure Page
1. Experimental phase diagram of iron ...... 4
2. Density of states of FCC c o p p e r ...... 29
3. Up spin energy bands of BCC iron at a lattice constant of 5.2 a.u ...... 37
4. Down spin energy bands of BCC iron at a lattice constant of 5.2 a.u ...... 38
5. Total density of state of BCC i r o n ...... 39
6. Up and down spin energy bands of BCC iron at a lattice constant of 9.0 a.u ...... 41
7. Up spin energy bands of FCC iron at a lattice constant of 6.5516 a.u ...... 43
8. Down spin energy bands of FCC iron at a lattice constant of 6.5516 a.u...... 44
9. Up spin energy bands of FCC iron at a lattice constant of 7.0 a.u ...... 45
10. Down spin energy bands of FCC iron at a lattice constant of 7.0 a.u ...... 46
11. Magnetic moment versus R for BCC and FCC i r o n ...... 59
12. Total density of state of FCC iron at a lattice constant of 6.5516 a.u ...... 61
13. Total density of state of FCC iron at a lattice constant of 6.8107 a.u ...... 62
14. Total density of state of FCC iron at a lattice constant of 7.0 a.u ...... 63
ix /eSTRACT
A survey of previous theoretical and experimental works on iron in some Bravais lattices and at various temperature and pressures shows this 3d transition metal to exhibit a wealth of allotropic transformations and to possess, along with some of its alloys, peculiar magnetic properties in the face centered cubic (FCC) phase. We present an ab-initio self-consistent spin polarized band structure study of this metal in the body (BCC) and face centered cubic lattices at absolute zero and at various atomic volumes. Our calculations employed gaussian basis functions in a LCAO scheme and used the RSK local density potential. In contrast to previous attempts, the structural and atomic volume dependences of the electronic and magnetic states of iron are analysed using the fundamental parameters, band widths and exchange splittings, of the energy bands.
While the smooth variation of the magnetic moment in BCC iron with lattice spacings greater or equal to 4.8 a.u. is attributed to the atomic origin of magnetism in that structural phase, its drastic change in FCC iron between a = 6.5516 a.u. and 7.0 a.u. is ascribed to a transition from an itinerant picture to a localized one. This analysis sheds some light on the nature of magnetism in other metals like cobalt. The relevance of the above transition to the properties of FCC iron based alloys is illustrated. Applications of the numerically established local quadratic dependence of the charge and spin form factors on the lattice spacing are discussed. CHAPTER 1 INTRODUCTION
The BCC phase of iron is known to be the most stable one in ordinary conditions of temperature and pressure. The Curie or critical temperature of this ferromagnet is 1043 K. The equilibrium lattice spacing at 0 K is around 5.4057 a.u. A quasi-exhaustive list of theoretical band structure calculations of iron (Fe) at the equilibrium lattice constant (slightly different for different 1 O calculations) is provided by Tawil and Callaway , Callaway and Wang , and recent reviews'*'4,5*®. Unlike the case of nickel, there is a fi ^ 7 ft Q general agreement between theory ' * and experiment ' ' ' for BCC iron as far as band widths and exchange splittings are concerned.
The experimental spin splitting obtained at P^ by Eastman et a !.5,
1.5 electron volts (eV) compares to the 1.34 eV theoretical result of
Callaway and Wang2. This is a significant agreement as the theoretical splittings for FCC nickel are generally double that of the experimental ones. The majority (or up) spin sub-band width at P is reported to be 3 and 2.6 eV by photo-emission experiment5 and p 7 theory , respectively. Danan and coworkers measured the magnetic moment to be 2.12 Bohr magneton (yg) while theory predicted 2.16
Eastman et a !.5 discussed the Fermi surface of iron and found theory2 and experiment to support one another.
The industrial importance of iron partly explains the continual growth of the number of experimental works on this 3d metal at finite temperature and under hydrostatic pressure*0'1* U n d e r the
l above conditions iron undergoes several allotropic transformations**»*4 and exhibit a variety of magnetic properties.
Liu and Bassett*4 presented an experimental temperature versus pressure (T-P) phase diagram*5,16 of iron. This diagram, whose
general features are reproduced in Figure 1, illustrates the regions of stability for the alpha (a), epsilon (e), gamma (y ), and delta (5)
iron. The respective crystal structures are the body centered cubic
(BCC), the hexagonal close packed (HCP), the face centered cubic
(FCC), and a BCC with magnetic disorder.
A major obstacle in the study of FCC iron, as intimated by
Figure 1, is the difficulty there is to stabilize it a low 11 17 temperature * . Several conclusions relevant to FCC iron are drawn from knowledge of various iron based alloys**. While there seems to be agreement on the paramagnetism of a high pressure HCP phase of iron, the question of magnetism in the FCC phase is yet to be totally settled. A high temperature FCC phase (i.e ., T = 1320 K) is known to 1 7 display spatial ferromagnetic correlations*Conflicting views exist on the magnetic state of the FCC phase assuming that it could be prepared at low temperature, and/or under high pressures. Kaufman et al.** discussed the Mossbauer effect, neutron diffraction experiments as well as susceptibility measurements for various FCC alloys containing iron. It was concluded that at low temperatures,
FCC iron would be anti ferromagnetic (AF) with a moment less than one
Bohr magneton (u^)* Two extrapolated lattice constants are obtained** for FCC iron at 25 degree celsius using the atomic 18 separation versus solute concentration curves of iron based alloys. The curves for Fe-Pt, Fe-Pd, Fe-Ni alloys in which iron 3
atoms possess a high moment (2.8 ^ ) lead to 6.878 a.u. while those
for Fe-C, Fe-Mn, Fe-N give 6.727 a.u. This was considered as
supporting Tauer and Weiss*^ hypothesis of low moment low volume (y^)
and high moment and high volume (y^,) states of FCC iron. The issue
of magnetic order in FCC Fe has been closely related to that of its 20 alloys in general and of the invar alloys in particular . These 21 invar alloys are characterized, among other things , by an almost
zero, if not negative, expansion coefficient in certain temperature
and solute concentration ranges.
In contrast to some of the above works where the properties of 22 FCC iron are infered from those of alloys, Wright , Gradmann _et_
al and Keune et al considered pure gamma iron films. Wright
studied by electron spin resonance 30 angstroms thick gamma Fe films
grown by electrodeposition on the (110) copper (Cu) single crystal face. He found these films to display ferromagnetism at room temperature. A moment of 1.1 uQ and a lattice spacing of 6.8027 were estimated. Gradmann et a l. performed magnetization measurements on gamma Fe thin films grown on the (111) copper surface. They utilized an oscillation type magnetometer. For temperatures in the range 70-
300 K a ferromagnetic ordering was observed in these films. Keune and coworkers carried out Mossbauer spectroscopy measurements on gamma Fe films. The films were grown on the (001) copper face and were reported to exhibit anti ferromagnetism for temperatures between 2d 2 and 295 K. These authors , considering the two previous results22»23, suggested a dependence of the magnetic ordering on the film orientation. 4
Figure 1. Major features of the temperature versus pressure
(T-P) experimental phase diagram of iron according to
Liu and B asset.^ The fu ll, dotted, and dashed curves
are respectively from references 14, 15, and 16.
2500 IRON
LIQUID 2000
P LJ gc 1 5 0 0
U
w 1000 v
500 a (B C C )
50 100 150 200 PRESSURE, kb 5
Gonser et a l.2^ presented Mossbaur spectra of coherent FCC iron
precipitates in a CuggAi^Q matrix. Contrary to the anti ferromagnetic
results of previous works on gamma Fe precipitates, as reviewed by
Kaufman et al.^ , these authors found ferromagnetic ones. They made measurements at 4.2, 77, and 295 K. They estimated an iron atom
separation of at least 5.0227 a.u. using Vegards Law.
Anti ferromagnetism and ferromagnetism were intimated to occur for atomic separations respectively below and above this value. A significant result is the observation of ferromagnetism in gamma Fe
by the microscopic Mossbauer technique as opposed to the macroscopic ones used in references 22 and 23. Very recently Brown et al >17, on the basis of their ferromagnetic results from neutron diffraction experiments on FCC iron at high temperatures, concluded that the low temperature FCC iron, if it could be stabilized, would be ferromagnetic.
Several other theoretical works on FCC iron exist in the
1 ite r a tu r e ^ - ^ . The APW band structure calculations of Wood^® were 91 followed by those of Deegan who used the narrow band approximation to the Korringa-Kohn-Rostoker (KKR) or the Green function method.
Dalton^® presented density of state curves for BCC and FCC iron along with other metals. 9 Q 30 More recently Madsen et al. , and Poulsen and coworkers'^ addressed, among other issues, the question of magnetic order in BCC and FCC iron for various lattice parameters. They employed a local density potential and the atomic sphere approximation (ASA) to the
KKR method. Their results and procedure of calculation are reviewed by Andersen et a l. ^ . For Wigner-Seitz radii (Rs) between 2.66 and 6
2.85 Madsen et a l. found the magnetic moment of an assumed FCC phase
to be between 3 and 3.5 yg. The moment drops rapidly below 1.5
around Rg = 2.60 and goes to zero at Rg = 2.47. Poulsen et a l.,
unlike Madsen and coworkers, included the sp-d hybridization in all
their calculatins and found no moment for FCC iron for Rs values less
than 2.60.
Kubler considered anti ferromagnetism which was not addressed
by the authors just mentioned. He also utilized a local density
potential to carry out self-consistent band structure calculations in
an augmented spherical wave approximation . He studied non-magnetic
ferro and anti ferromagnetic orderings in BCC and FCC iron. From the
total energy curves he found the stable magnetic order to be ferro
and anti ferromagnetic for BCC and FCC iron phases, respectively. A
difficulty connected with the FCC phase consists in having its total
energy lower than that of ferromagnetic BCC iron for the Rs values
considered. We shall discuss this subsequently. The moment of the
ferromagnetic FCC phase was found to exhibit the sharp drop described
above around Rfi = 2.63 and goes to zero in the vicinity of R$ = 2.55 a.u.
Roy and Pettifor3^ reported to have obtained the and ^
states from Stoner theory33. The ASA and local density
calculations of these authors predicted both states to be ferromagnetic. The experimental T-P diagram of iron in Figure 1 has
very recently been qualitatively reproduced by Hasegawa and
Pettifor36. Their calculations were based on the spin fluctuation 37 3ft theory of band magnetism * . They found the magnetic contribution to the free energy to be the driving force in the phase transitions and questioned previous explanations** which employed the two gamma
states.
From this survey it appears that the magnetic state of FCC (y)
iron at low temperature is yet to be determined as there are experimental and theoretical results to support opposite views.
Our goal is to study the problem of the stability of the magnetic structures by means of ab-initio self consistent electronic band structures of ferromagnetic iron in BCC and FCC Bravais lattices for some Rg values between 2 and 5. Paramagnetism is indirectly allowed for. A close analysis of first principle energy bands and
related fundamental parameters, band widths and splittings, is intended to explain essential electronic and magnetic properties of iron in the above phases as the lattice constant changes. The structural dependence of magnetism in iron is not only of importance in its own right but also is expected to be relevant to a complete understanding of invar anomalies.
The second chapter of this dissertation is devoted to an outline of our method of band structure calculation, the choice of a potential and the formalism for the calculation of the total energy of a metal recently developed by this group. The contraction procedure and other computational details are described in chapter two. The analysis of the results follows in chapter three and four. The electronic structure of BCC and FCC Fe are presented in the former while the latter addresses the magnetic properties. The discussions in the last chapter include some suggestions for future investigations believed to be of relevance to the properties of transition ferromagnets and a summary. CHAPTER 2 GENERAL THEORY
A modern treatment of energy band theory for a metal generally amounts to solving the Kohn-Sham equations J of density functional theory3^*4®’4*. Indeed, assuming a potential describing the many particle system is known3^*4^ there exist several methods of obtaining the energy levels. These band calculation procedures have been reviewed by Callaway43.
In the first section of this chapter we shall outline the linear combination of atomic orbital (LCAO) version of the tight-binding44 method. Our choice of an effective one particle density functional potential is discussed in the second section. A new formalism for the calculation of the total energy of a metal is then covered in the last one.
A. The LCAO Method
Our aim is to solve the single particle Schrodinger equation:
( - V 2 + vo (?)) (f,f) = En(ic) *n (it,?) (1)
where 9 and
*n$»r) = e v *n (k,r + ?u), (3)
satisfies Bloch theorem. The band index n, the coordinate vector r, and the position vector for lattice site u are defined as usual.
The potential V (r ) comprises coulomb terms and an exchange correlation contribution V „ (r) where a indicates the spin. The xca r nuclear charge is Z and the units of energy are Rydbergs (Ry) with
0 = c = 1, e2 = 2, m * 1/2.
The tight-binding method consists in searching for solutions to equation 1 by expanding the wave function >?n(i<,r) in a finite set of atomic like (i.e., localized) functions. When the radial parts of these functions are taken to be those obtained for atoms the method is referred to as the LCAO method. The term LCGO, for linear combination of gaussian orbitals, is used if the above radial parts are gaussians. The basis functions
*.(£,?) = N’1/2 E e ^ U.(r - Ry) (4)
with
Ufn (6,*) (5)
in which K (0,i|») and Rfl (r) , respectively, contain the angular X.. 9m and radial dependences. The orbital quantum number is and m is the magnetic quantum number when the K (o,^) are taken to be 10
spherical harmonics. In general the choices of R (r) are i
-a.r A- -a.r2 Rt< = N.r 1 e 1 or R^ = N.r 1 e 1 (6)
where is a normalization constant. For treatment of cubic
crystals the angular functions are often chosen to be cubic
harmonics4^. The substitution of
y i< ,r ) = I Cni(ic) t . t f . r ) (7)
into equation 1 leads to
where
-ii<*$ HiJ (ic) = j e u < U.(r - | -V2 + Vo(r) | \).(r) > (9)
and the overlap matrix elements are
S. .(k) = S e y < U.(r - | Uj(r) > (10)
The matrix equation 8 is solved for the eigenvalues (ic) and the expansion coefficients C .(ic) . m It appears from equation 8 that the mathematical foundation of the LCAO method is the Rayleigh-Ritz variational formalism. In the absence of orthogonality between the basis functions, as is the case 11
here, the overlap matrix elements, S. .{it) , appear explicitly. * J To construct the potential Vo(r) one needs the charge density
p(r) which is generally obtained only when the correct wave
functions of the occupied states of the system are known. For this
reason, the most recent applications of the LCAO method involved an
iterative procedure. The reader is referred to Wang and Callaway4^
for details on the iterations. A second major problem in energy band
theory, besides finding a method of solution to equation 1, is the
choice of an effective potential as we shall discuss in the following
section.
B. The RSK Local Density Functional Potential
The Slater exchange potential, proportional to the 1/3 power of
the charge density, is a well known example of an effective, local
and single particle exchange potential. Hohenberg and Kohn41 proved
that the ground state energy of a many electron system is a unique
functional of its ground state charge density. Based on this
theorem, Kohn and Sham'*® and Sham and Kohn41^ derived the equations
describing real systems like metals. This led to the constructions
of various exchange-correlation potentials, referred to as density
functional potentials, which differ in their estimations of the
unknown exact exchange-correlation contribution to the unique functional. One of the most used form is due to Von-Barth and Hedin
(VBH)4'*, a more soundly based version of which has been presented by
Rajagopal, Singhal and Kimball (RSK)46. This RSK potential has been fitted to the VBH functional forms, but possesses slightly different parameters. 12
The exchange or exchange-correlation potentials to be discussed
depend on the spin, o. This leads to a splitting of equation 1 into
two, one for each spin. For ferromagnetic systems this leads to
differences between eigenvalues En+(£) and En_(^) » 1*e., the
exchange splittings.
Following Von Barth and Hedin, the exchange-correlation
potential is written:
V*C0 * ' " C»
p(r) = p+(r) + p_(r), exc(p+(r),p_(r)) = ex + ec (12)
with
P+(r) + P_(r) e = 3 vl/3 x ■3 <*r> (13)
ec == £l + {£l ’ £c ) f(x)*
For
P+(r ^ p (r) x+ * x = (14) P(r) p(r) and aQ = 2"*1/3 one has
(15) f(X) ■ - r r h - <*+4/3 + *4-/3 - V 0 13
The quantities e£ and e£ are expressed in terms of four constant parameters:
cP = 0.0504, rP = 30, cf = 0.0254, rf = 75 for VBH (16)
potential while
cp = 0.04612, rP = 39.7, cf = 0.02628, rf = 70.6 (17)
for RSK potential.
The merits of local density potentials are discussed in several review articles4^* ^'48. In particular, as discussed by Callaway, the VBH type potentials give quite reasonable values of the band widths for metals while a typical Hartree-Fock treatment yields widths which are double that of the experimental ones. The recent work on lithium ^*50 demonstrated that the RSK potential correctly describe the limit behavior of this metal for lattice constant larger than normal. In contrast to some Hartree-Fock results, the energy levels of metallic lithium unambiguously go to the atomic ones for lattice constants larger than about 14 a.u.
As for possible limitation of a local density potential as compared to non-local ones, the very recent study by Wang81 is a case in point. Wang performed self-consistent band structure calculations for FCC nickel utilizing a non-local density functional potential.
This potential comprises a non-local exchange part constructed CO according to Gunnarson et a l. weighted density approximation scheme and a local correlation contribution.88 The resulting exchange correlation energy per particle, contrary to the case of local approximations, goes to 1/r at large distance. It therefore formally reproduces the correct behavior away from a point charge. However 14 the results of Wang calculations, as compared to those obtained^ with the local \©H potential have mixed behaviors. Indeed, while the observed decrease of the exchange splitting from 0.6 eV to 0.4 eV was desired, the somewhat increase obtained for the band width was not.
A generally shared contention at present is that non-local forms of density functional potentialhave yet to prove themselves to be superior to the local ones in practical applications. Perdew and coworkers^ obtained some results in that direction for atoms and other systems but the case of metals is not yet resolved.
In a LCAO scheme and using any VBH type potential we shall derive in the next section a formalism for the calculation of the total energy of a metal.
C. Evaluation of the Total Energy of a Metal
Callaway et a l.49,5^ recently presented a formalism for the evaluation of the total energy of a metal in the framework of an LCGO band structure calculation. After expressing the kinetic energy in terms of eigenvalues Ena$ ) one can write the total energy of a solid as:
N Et = I E (it) - / i-d l-Plf-H d3rdV + £ ----- T no !?-?•! tfXv l i u g
+ Exc - § I °o(f> Vxco'f > d3ri (18) where the sum over n and a is limited to occupied states. For a single specy system as metallic lithium = Zy * Z, the nuclear charge. The explicit exchange-correlation terms in equation 18 are 15 combined into A xc
where
Wa(r) = p(r) ( 20)
It is to be noted that Exc is the exchange correlation contribution to the total energy not A . A major difficulty overcome by this XC group is that of evaluating the infinite Coulomb terms. These terms have been rearranged so as to obtain finite quantities U and D by setti ng
U + D = - / P-Cf- 'J — d3r d3r 1 + E |r - r*| t$ with
( 2 1 )
and
( 22)
For reasons to become clear in the following chapter, U and
D are evaluated in reciprocal space.
u - - £ N a E VT (*s ) Pp(i and D = \ N Z [ (-|) C - VT(0) - S| Q Ve(lts )]. (24) In these expressions, restricted to a monotomic system, 0 is the cell volume and the index e denotes the electronic contribution. Vj(Kg) is the Fourier transform of the total Coulomb potential, including the nuclear-nuclear repulsion, and Ks is a reciprocal lattice vector. Vj(0) is described by Callaway^ and the constant C: c = slo “T? ' f “hr d3q* (25) K Q S H has been evaluated by Pack et al.5^ for various lattice structures. Following these authors' approach, this constant has been reevaluated 49 to more significant figures in the course of the work on lithium . _ _ 2 Boettcher went farther and found, in units of : a C = -8.913632917586 for simple cubic lattice C = -11.432989069675 for body centered cubic (26) C = -14.403769009759 for face centered cubic. The calculation of is basically that of W (r) which, after some xc o' algebra, can be written W0 (r) = [\ uP + vc + 6) (2 x o )1 /3 + Q - G (27) with 17 The parameters in the expressions of W are as defined in reference 42 and aQ = 2“^ ^ . The total energy is thus obtained as: N ET = £ E (£) + U + D - a . (28) T no no' ' xc ' ' Wang and Call away^ implemented the LGCO method of band structure calculation using Slater type or VBH potentials. The resulting program package, BNDPKG, has been extended to evaluate the total energy 49,50^ -phis program package has been used in the present study of iron, and the following chapter is devoted to the salient features of the computations. CHAPTER 3 COMPUTATIONAL PROCEDURE By computational procedure, what is meant here is the set of numerical techniques and details pertaining to the implementation of the formalisms in the preceding chapter. The major steps of the calculation are described, followed in the second section by the discussion of the contractions of basis sets. A. BNOPKG In a single particle approximation the band package, BNDPKG, produces the energy bands of a cubic material with one atom per unit cell. The calculations are non-relativistic and at absolute zero temperature. Para- or ferromagnetic orderings are allowed for and the LCGO method is chosen. The angular functions are cubic harmonics. The eigenvalues and wave functions are obtained at as many number of £ points in l/48th of the first Brillouin zone as desired or economically feasible. Emphasis, in what follows, will be put on general features and new ones pertaining to the evaluation of the total energy. For further details we refer to Wang and Callaway45 and appendix A where some extended programs are provided. FCOF is the first program of the package. It employs gaussians orbitals provided by atomic calculations to construct an initial charge density and then generates the fourier coefficients of the various terms in the potential, i.e., the electronic and total Coulomb and exchange-correlation potentials. The innocent looking 18 19 matrix elements, H. .(it), contain three dimensional multi-center 1 J CO integrals. As shown by Lafon and Lin , the use of Fourier expansion, i.e. V ? ) = jE V * s > cos t s- r, (29) reduces the difficulties in the evaluation to at most two-center integrals. In the above expressions, the use of the cosine function is justified by the inversion symmetry of the potentials. The resulting two center integrals can be evaluated analytically if gaussian functions are used^*^®*^. An Ewald split technique is used to remedy a possible slow convergence of Fourier sums due to the screening potential of tightly bound electrons and the contribution of the nuclear terms. The choice of an effective exchange-correlation potential is determined by the input to subroutine VXCRS. Slater or VBH type potentials are currently allowed for. The new version of FCOF produces the Fourier coefficients of W Cf(?) needed for the total energy calculation and contains a new approach to the computation of Vj(0). This quantity has little bearing on the relative position of the energy levels but strongly affects the total energy as can be seen from equations 24 and 28. Program ESINT evaluates integrals needed in the Hamiltonian and overlaps matrices and BND forms, using properties of the cubic group, the appropriate combinations to generate the matrix elements. In the event where self-consistency is not needed, the diagonalization option in BND is chosen to obtain the energies and wave functions. 20 We recall that only ft-points in the irreducible wedge (1/48^ of the first Brillouin zone) need be considered for cubic crystals. For self-consistency procedure the necessary Fourier transforms of the overlap integrals are generated and properly arranged in the fourth (SIJ) and fifth (INVSIJ) programs respectively. The iterations toward self consistency are performed in SCF1 and SCF2 for paramagnetic and ferromagnetic calculations respectively. Changes to the Fourier coefficients of the charge density, Coulomb and exchange-correlations potentials, are obtained and added to the appropriate quantities. Self-consistency is assumed to be reached when the Fourier coefficients of the Coulomb potentials from two consecutive iterations differ by no more than 10"^ for any it& vector. A judicious choice of the damping parameter FACT (i.e ., 0.2 - 0.4) reduces the number of necessary iterations. SCF1 AND SCF2 has been extended to obtain the self-consistent Fourier transforms of W or respectively. The new version of SCF2 play a key role in the forthcoming discussion of the magnetic properties of iron. Indeed, in the course of the study of metallic lith iu m ^* ^ the total energy of this material was computed using both para- and ferromagnetic inputs. For lattice constants less than 10.5 a.u. the paramagnetic calculations yielded energy bands identical to the up and down spin sub-bands of the ferromagnetic results and the two total energies agreed to over 5 decimal places. On the contrary, for lattice parameters above 10.7 a.u., lithium was found to be ferromagnetic. In this range the total energies for the ferromagnetic calculations were clearly lower than those resulting from the paramagnetic ones^0. These results did not depend on the 21 input magnetic moment which was chosen between 0 and 1. Similar results have been obtained for copper. The important point here is that ferromagnetic calculations for iron, at various atomic volumes, are expected to yield a paramagnetic result if indeed iron is paramagnetic at the lattice constants considered. The self-consistent results are fed into ESINT which outputs are used by BND to calculate the final self-consistent band structure. The last program of the package, DENST, generates the electronic density of states. This program has been extensively modified to evaluate the total energy expression in equation 28. Bagayoko et al. ^ used BNDPKG to obtain the electronic energies and other quantities for FCC copper at a lattice constant a = 6,80915 a.u. A basis set of 75 gaussian functions®^ has been used and it took 417 CPU minutes (on an IBM 3033) to obtain self-consistent eigenvalues at 505 It points of the irreducible wedge. The computations employed a paramagnetic input. Similar calculations with ferromagnetic inputs take about twice as much time. This makes it financially intractable to study BCC and FCC iron, with ferromagnetic inputs, at several lattice constants unless a method is found to significantly reduce the time needed. The contraction of the basis set is one such procedure as is discussed in the next section. B. Contractions Despite its advantages, i.e. analytical evaluation of integrals, A -a r the use of gaussian ( r e ) basis sets has a drawback. Namely, the size of the Hamiltonian matrix is about twice as much as it would 22 A . - a . r be with exponential basis functions ( r e ). Contraction schemes has been developed 63-69 in order to reduce the dimension Nd of the Hamiltonian matrices for atomic and molecular calculations. Two methods generally utilized are the least square fit®®*®® and the formation of linear combinations with fixed coefficients, of single Gaussian type orbital (GT0)6:**66»69. The basis functions resulting from the latter will be referred to as contracted Gaussian type orbitals (CGTO), not to be confused with Gaussian type orbitals obtained through a scaling of the exponents^®. Bagayoko^ presented guidelines for the construction of CGTO suitable for solid state computations. Those are only surveyed here as further details are available in reference 71. The sets of GTO or CGTO provided for atoms are generally not sufficient for molecular or solid state calculations. A major reason for this is the difference between the electronic distributions in atoms, molecules and solids. While the bond formation characterizes molecules69, the quasi-free electron behavior distinguish metals from the latters and from atoms. One solution to the above inadequacy of 17 7^ atomic GTO basis is the inclusion of supplementary orbitals * . In particular the atomic GTO must be supplemented by a 4p orbital in order to correctly reproduce the electronic energy bands of 3d metals around the fermi energy. A specific example is the disappearance of copper fermi surface neck at L, as shown by Bagayoko^, if a 4p like orbital is not used. Another, relatively minor, modification of the atomic GTO sets is the dropping of the atomic s orbital with the smallest exponent for calculations on 3d metals. The similarity of the core electronic cloud in atoms, molecules, 23 and solid supplemented by the fact that the largest number of orbitals, for a given symmetry (s, p, d, or f), contribute to the innermost atomic functions lead to an obvious first rule to follow in forming CGTO. It consists in grouping orbitals with consecutive exponents so that the number of single GTO involved in a CGTO decreases with decreasing exponents as shown in Table 1, relaxing the contraction of the outer orbitals in going from atoms to molecules and solids in that order. To facilitate the forthcoming discussion, this prescription will be referred to as bond or free electron rule for molecules or metals respectively. The constant factors multiplying each orbital present in a CGTO remain to be specified. The atomic calculations provide 1, 2, and 4 sets of expansion coefficients for the d, p, and s orbitals of copper respectively. In the absence of further exponent optimization in forming the CGTO one employs the atomic coefficients. The question is then the choice of the sets of coefficients to use. Given the exponents pertaining to the s symmetry, one has to choose among the expansion coefficients for the Is, 2s, 3s, or 4s atomic states to form the CGTO describing the s symmetry functions. Basch et al found the Is atomic coefficients to describe all s orbitals far better than do the 3s coefficients. They mentioned a similar result reported by Huzinaga. Their finding has been verified for copper and nickel. This result could be expected, considering that the largest contribution to the total energy for a given symmetry is that belonging to the lowest principal quantum number, i.e ., Is for s, 2p for p, and 3d for d symmetries. This conclusion will be referred to as the lowest n rule. In the following all the CGTO have been 24 constructed using the Is, 2p, 3d atomic coefficients for the s, p, and d orbitals. If one utilizes the atomic coefficients as described above then elementary mathematics shows that grouping together two orbitals with expansion coefficients of different signs and exponents not far apart, is almost equivalent to eliminating them both as far as the charge density is concerned. The sign rule simply consists in avoiding such combinations. In the Rayleigh-Ritz variational formalism the m 1'*1 variational Nd eigenvalue, X where Nd is the number of basis functions, tends to m the true m^ eigenvalue when Nd increases. Therefore, energy levels obtained with CGTO (reduced basis set) will generally not be lower than those generated using single GTO. A consequence of this is the upward shift of the energy levels to be expected when employing CGTO. This may appear as a drawback of the use of CGTO but should be of no effects on relative quantities such as exchange splittings and band widths resulting from solid state calculations. The program package BNDPKG has been run to produce the non self- consistent energy levels of copper at high symmetry points of the Brillouin zone. The input for subroutine GTO determine the choice of contraction. The criterion for best contraction is the lowering of the energy levels. The above discussion of the limit behavior of variational eigenvalues as Nd increases makes it difficult to compare two contractions employing different numbers of CGTO. The 18 contractions in Table I, all of which involved 32 CGTO, has been used in these test calculations for copper. They are ranked in Table II from the best to the worst. One difficulty in choosing the overall TABLE I. Contractions,eighteen different contractions (Nd = 32) used in the non self-consistent band calculations. The GTO utilized are those provided by Wachters. For a given synmetry the orbitals are numbered in the order of decreasing exponents. Contrac tions S orbitals p orbitals D orbitals 1 (1,2,3,4,5.6) (7,8) (9) (10) (11) (12) (11) (I,2,3,4,5.6) (7) (8) (9) (10) (1,2,3.4) (5) 2 (1,2,3.4,5,6,7) (8) (9) (10) (11) (12) (13) (1,2,3,4,5,6) (7) (8) (9) (10) (1,2,3,4) (5) 3 " " " " " " " (1 ,2 ,3 ) (4,5 ) 4 (1,2,3,4,5,6) (7) (8,9) (10) (11) (12) (13) 5 (1,2,3,4,5,6) (7) (8) (9,10) (11) (12) (13) 6 (1,2,3,4,5,6) (7,8) (9) (10) (11) (12) (13) (1,2,3.4,5) (6,7) (8) (9) (10) (1,2) (3.4,5) 7 ■ " " '* " (1.2,3) (4,5) (6,7) (8,9) (10) (1,2,3,4) (5) * " " ” H (1.2,3) (4,5) (6,7) (8) (9,10) (1,2,3) (4,5) 9 " " " " (1,2,3,4,5) (6,7) (8) (8) (10) 10 " " " " (1,2,3,4,5) (6) (7) (8,9) (10) (1,2,3,4) (5) 11 " " " " (1,2,3,4,5) (6) (7,8) (9) (10) (1,2,3) (4,5) 12 " " " " (1,2,3,4) (5,6) (7) (8,9) (10) (1,2,3) (4,5) 13 " " " " (1,2,3,4,5) (6) (7) (8,9) (10) (1,2) (3,4,5) 14 " " " " " " " (1,2 ,3 ) (4,5) 15 .. » .. » ,. „ (l) 2 (>3t*i5) 14 " " " " : (1,2,3) (4,5) (6,7) (8,9) (10) (1,2,3) (4.5) 17 " " " ! " » (1,2) (3,4,5 ) 18 " " " " ** " " (1) (2,3,4,5) in TABLE II. Comparison of different contractions.The contractions are those in table I and are ranked from best to worst. The best is the one giving the lowest eigenvalues. The rankings change with symmetry points. S ta te s C o n tra c tio n Numbers A max = max(AW- A**) m m 1,4,5,2,3,8,9,6,11,10,14,12,13.15,7,16,17,18 +3.053 Ryd “\p 1,7,10,8,9,11,12,14,16,6,13,15,17,18,2,3,4,5 +1.035 a f r 1,2,7,10,6,13,17,8,9,11,12,14,16,15,18,4,5,3 + 0.011 1,2,3,4,6,8,9,5,11,7,10,13,17,16,14,15,18,12 +0.166 i f L 1,2,3,4,5,6,8,9,11,10,7,12,14,13,15,16,17,18 +0.120 |1 X 1,2,3,4,5,8,9,6,11,10,7,12,14,16,13,17,15,18 +0.111 Vw 1,2,3,4,5,6,8,11,9,10,12,13,14,15,7,16,17,18 + 0.122 Higher Bands 2,3,1,4,5,9.11,10,14,12,6,13,8,7,16,17,15,18 27 best contraction is the change of the ranking from one state to another. One refers to the system specificities in order to make a choice. The electrical, thermal and optical properties of a metal being primarily determined by the valence and higher bands these levels will be considered the most important for copper. Let A*"* and A* designate the m^ eigenvalues obtained with the best and m m 3 worst contractions, respectively. At the r point there exists 6 eigenvalues corresponding to the valence states (degeneracy included) of copper. The quantity A = max (*; - ), (30) max 'mm'* ' ' where m takes 6 different values, has been used to compare the contractions. The values of Amau in column 3 of Table II refer to IllaX the comparison of the best and worst contractions which are respectively the 1st and last ones as they appear in a row of Table II. The detailed discussion of the contents of Table I and II is available in reference 71. With the knowledge that the twelfth Is and the eight 2p orbitals have negative expansion coefficients the verification of the guidelines set above become apparent from these tables. Contractions 1-2 and to some extent 3-4, are the best for valence and higher bands while 15-18 are the worst for the same states. The best ones follow the guidelines best while the worst ones violate one or more of the rules. Interestingly, except for the deletion of the s tail and the inclusion of a tenth p orbital, contraction 14 corresponds to the best obtained by Wachters 63 for 28 atomic copper. Contractions 1, 2, 3, and 4 are in this order, except at r , for the valence states. This means, A being relatively maX small at r, that either one is expected to give good results. The best contraction in Table II (contraction 1) has been chosen to perform self-consistent band structure calculations for FCC copper. At the exception of the contraction of the basis set every other parameter entering these computations has been kept as it was in the work of Bagayoko et a l.6^ where 75 GTO have constituted the basis set. Let A32 and A7^ be the mth eigenvalues, beginning with the Is state, resulting from the former and latter calculations. A detailed comparison of the valence bands for the high symmetry points, show: A75 < A32 E75 < E32 m m » l F < F and 32 32 ^75 - eJ5 - X 0.0025 Ry, (31) m m - ef * In the above expressions the Fermi energies 75 32 75 32 Ec and E,. as well as A and A are negative quantities. This r r m m 32 relation therefore shows that while the A has been shifted upward m r 75 as compared to the corresponding Am , relative quantities like the occupied band widths have practically not been changed. This is the rigid upward shift (0.0142 Ry), due to the use of small basis set expected on variational grounds. The shift is illustrated by figure iue . est o sae NE fr aaantc C cpe » FCC copper paramagnetic for N(E) states of Density 2. Figure STATES/ATOM - RYDBERG 104 40 24 46 32 64 60 72 56 88 96 * 1.4 upward shift of the bands for Nd = 32 is seen in the the in seen is 32 Nd = for bands the of shift upward rigid of The GTO bases 32 CGTO and using 75 and respectively. .obtained Rydberg per states/atom in hit o h rgt f h Fri ee ad h peaks the and level Fermi the of right the to ift sh n h dniy f tts uv . = .01 a.u. 6.80915 = a . curve states of density the in - 1.2 - 1.0 ENERGY (RYDBERG) - 0.8 - 0.6 - 0.2 -00 29 30 2 where the density of states resulting from both calculations are shown. Another test of the guidelines for contraction is provided by the total energy calculation re s u lts^ . Not only a minimum is obtained in the total energy curve, but also it occurs around a lattice constant of 6.80915 a.u. which is almost identical to the £ experimental value for the equilibrium spacing at absolute zero. The above results on a 3d metal like copper leave no doubt about the adequacy of the above contraction procedure for the study of other 3d materials or lighter ones. The self-consistent calculations with 32 CGTO and 75 single GTO have been performed on an IBM 3033. The total CPU time required for the former was 6.8 times smaller than that (417 minutes) needed for the latter. This is the significant reduction of CPU time we were aiming for. Other miscellaneous benefits of this procedure are described in reference 71. In particular we have explicitly shown the inadequacy of some contractions for atoms and molecules in solid state calculations. BNDPKG has been used with a contraction option, following the above guidelines, to extensively study iron at 0°K. The results of those calculations are presented in the next chapter. CHAPTER 4 ELECTRONIC STRUCTURES OF BCC AND FCC IRON The new version of the band package, BNDPKG, has been used in this work. We employed the RSK local density potential. Choosing the contraction option of BNKPKG we constructed 8 s,5p, and 4d CGTO according to the guidelines set forth in the preceding chapter. The atomic GTO entering the CGTO were those provided by Wachters . Only ferromagnetic inputs were considered and the ab-initio self- consistent band energies have been produced at 505 and 506 £ points of the respective irreducible wedges of the FCC and BCC structures. The input configurations, for both structures, were 3d^4s*. These calculations have been carried out for several values of the atomic volume. The electronic structure of BCC iron at a = 5.4057 a.u. will be discussed in section A, followed by the studies of the variations of the bands with atomic volume for BCC and FCC lattices in sections B and C. The last section is devoted to the charge and spin form factors. A. BCC Iron, a = 5.4057 a.u. As previously stated most of the previous calculations for BCC iron at normal lattice constant (i.e ., a = 5.4057 a.u., Rg = 2.6616 a.u.) are listed in the aforementioned references 1_6. A comparison p of our results to those of Callaway and Wang is particularly meaningful as the differences between the two works reside in our use of 43 CGTO and the RSK potential instead of 75 GTO and the VBH 31 32 potential. The present calculations basically reproduced the results of Callaway and Wang. The present eigenenergies are rigidly shifted upwards as expected on variational grounds. However, in this case, contrary to the work on copper described in Chapter 3, the shift is primarily due to the use of a different method of evaluating Vj{ 0 ). This Fourier coeffficient of the Coulomb potential, as previously stated, does not affect the relative position of the bands. The algebraic values of the differences between the present and previous V-j-(0) and Fermi energies (Ep) are respectively 0.4376 and 0.4402 Ry. On the scales of the graphs in reference 2, the band structures these authors obtained are indistinguishable from ours if the Fermi levels are superposed. Tables III and IV contain major quantities describing these bands. The magnetic moment, 2.15 pB, is 0.01 ^ less than the value they found. The up spin d band width at p is 2.5 eV, 0.1 eV less than their result. The exchange spliting at P 4, 1.31 eV, is 0.03 smaller than the previous value2. We found charge and spin form factors almost identical to those obtained by Callaway and Wang. The analysis by Van Laar et al.^ therefore shows than the spherical parts of both spin form factors are in accord with experiment. The above agreements between our results and previous theoretical as well as experimental ones justify the application of a contracted basis set to the study of iron at several atomic volumes. The slight reductions in the magnetic moment and the exchange splitting seem to indicate a somewhat stronger correlation effect in the RSK potential as compared to the VBH. Further discussion of the electronic structure of iron at normal lattice constant is presented in 33 conjunction with that of the variations of the bands with the lattice parameter. Table III. Representative band widths (H) and exchange splittings Wp* Rs a W sPl wd "d Wd(4i) 6Ee x ,ri> {Eex(Nl> fiEex lP4> SEex,P3» (Hl 5 r r i 4 » 2.4619 5.0 1.5533 0.2579 0.3262 0.4017 0.0176 0.0691 0.0755 0.1438 2.5603 5.2 1.4326 0.2165 0.2913 0.3798 0.0159 0.0794 0.0885 0.1632 2.6616 5.4057 1.3211 0.1822 0.2579 0.3540 0.0150 0.0855 0.0961 0.1718 2.7573 5.6 1.2269 0.1557 0.2310 0.3349 0.0147 0.0916 0.1039 0.1793 2.9542 6.0 1.0626 0.1114 0.1857 0.3094 0.0157 0.1079 0.1237 0.1980 3.4466 7.0 0.7742 0.0425 0.1141 0.2758 0.0212 0.1417 0.1617 0.2332 4.4313 9.0 0.3808 0.0062 0.0617 0.2792 0.0533 0.1881 0.2176 0.2534 Table IV. Representative band widths (W) and exchange splittings ( fiEex) for BCC-iron - in Rydbergs (Ry) - for various atomic radii (R ) or lattice constants (a). W°cc is the 5 d occupied d band width. Up or down spin is indicated by t or +, respectively. R a WOCC wH+ s Wd d waH+ 6Ee x (H12> 6Ee x (H2 5 ') *H12 + ”H25 ' *H12+" H25'+} (a.u .) ( a .u .) (Ry) (Ry) (Ry) (Ry) (Ry) 2.4619 5.0 0.4602 0.5003 0.5287 0.0900 0.1183 2.5603 5.2 0.4031 0.4159 0.4498 0.1054 0.1393 2.6616 5.4057 0.3507 0.3435 0.3764 0.1169 0.1499 2.7573 5.6 0.3091 0.2844 0.3151 0.1292 0.1599 2.9542 6.0 0.2732 0.1838 0.2061 0.1612 0.1835 3.4466 7.0 0.2421 0.0546 0.0419 0.2243 0.2116 4.4313 9.0 0.2266 0.0002 0.0496 0.2729 0.2235 36 B. Volume Dependence of BCC Iron Band Structure The up and down spin sub-bands of BCC iron at a = 5.2 a.u. in Figures 3 and 4 are shown to facilitate the forthcoming discussion. Figure 5 shows the total density of states. Fundamental parameters describing the bands, widths and exchange splittings, are provided in Tables III and IV. Appendix B contains the valence state energy levels, at high symmetry points, for a = 5.2, 5.4057, and 5.6 a.u. The band widths decrease with increasing lattice constant and more up or down spin bands are respectively filled or emptied. These patterns can be described in terms of a competition between the kinetic and the exchange energies which get respectively smaller and larger in magnitude as the lattice constant increases. One drastic change in the band structure of ferromagnetic BCC iron occurs at the N point. For a = 5.2 a.u., Fig. 3, the upper and lower Dg branches are connected to Nj' which lies below Ng where Gg and the upper Dg meet for the up spin sub-bands. In the down spin case, Fig. 4, Gg and the upper Dg meet at Ng which lies above N4 where the upper G4 and Dg are branching off. For the minority sub-bands this picture remains the same up to a = 7.0 a.u. where the upper Nj, N4 , and Ng are in that order of increasing energy but are almost degenerate (E(N^) - E(N^) = 6.9 rnRy; E(Ng) - EfN^) = 5.6 mRy). For the majority spin sub bands N^' and Ng become degenerate around a = 5.4057 (see Fig. 1 of Callaway and Wang^, RS * 2.6616). This degeneracy no longer exists at a = 5.6 a.u. and above and the Gg branch is connected to Dg at Ng which has moved below Nj' where the upper G4 and Dg meet. ENERGY (RY) some lines along BCC iron ferromagnetic in bands Energy 3. Figure f ih ymty o te aoiy r p pn Energies spin. up or majority the symmetry for high of r i RdegR) Te atc cntn i a 52 a.u. 5.2 = a is constant lattice The Rydberg(Ry). in are A HGNDPA T2NP FH 37 ENERGY (RY) Figure 4. Energy bands in ferromagnetic BCC iron along some lines some lines BCC along iron ferromagnetic in bands Energy 4. Figure - - - - 0.8 0.4 0.4 0.0 0.6 0.2 0.2 r i RdegR) Te atc cntn i a 52 a.u. 5.2 = a is constant lattice The Rydberg(Ry). in are of high symmetry for the minority or down spin.Energies down spin.Energies or minority the symmetry for high of '25 25 N '25 38 39 Figure 5. Total density of states N{E),in states/atom per Rydberg, and the valence electron number for ferromagnetic BCC iron. The lattice constant is a = 5.2 a.u. 56 48 in z i- 32 o EC I- o LJ _l UJ 0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 ENERGY( RY) 40 The passage of the Ng representation below 1 in the majority spin sub-band is a direct consequence of the above mentioned competitions between kinetic and exchange energies. The d states (Ng) are far more sensitive to the exchange effect than are the P ones (N^1). When the lattice constant increases so does the exchange effect and Ng, for the up spin sub-bands, passes below (N^‘ ). There follows a change of connections to avoid the crossing of bands of the same symmetry, i.e ., the upper and lower Dg. A second major change in the structure of the d bands is observed at a value of the lattice constant around 9 a.u. Figure 6 shows the up and down spin sub-bands for this lattice parameter at which the crystal field splitting is larger than the overlap effects; t 2g and eg sub-bands are consequently sp lit. It is apparent from figure 6 that the exchange splitting of the d bands is then much larger than sub-band widths. In other words, the crystal field splitting is larger than the overlap effect and t 2g and eg sub-bands are consequently sp lit. The values of the exchange splittings and band widths in Table III and IV, deserve comments. Callaway and Wang reported a rather large difference between the splittings at P 4 and Pg. This result has been reproduced here for a = 5.4057 as discussed in Section A of this chapter. One further observes, from Table III, that the increase of lattice constant is accompanied by a decrease of the ratio of the splitting at P 3 to that at P 4 . For a = 5.2 a.u. this ratio is about 2; it tends to 1 as the lattice spacing increases because both splitting approach the same value which is that for the atoms. While the total d band width at P, approaches that same atomic splitting, the widths for up and down spin sub-band go to zero. A known measure of the Figure 6 . Up (full) and down (dashed) spin energy bands of BCC iron at a lattice constant of 9.0 a.u. The following representations are in the order of increasing energy for both sub-bands: r25«* ri2* ^25' > H-|2> Pg* 0.0 0.2 0.6 0.8 42 hopping time for an electron occupying a given band is h/w where h is the Plank constant and W is the band width. The smaller W, the more time the electron will spend on a given site. The relatively small d band widths as compared to the sp band widths is a result of the relative localization of the d electrons as compared to sp electrons. A major characteristic of the BCC band structure of iron is the quasi-flatness of the D^, D2, upper and branches at lattice parameters equal or greater than 4.8 a.u. To this flat portion in the band corresponds a sharp peak in the density of state curve as shown in Fig. 5. These branches do not participate to the fermi surface. They therefore seem to possess a highly localized character as we shall discuss in connection with the study of the magnetic properties. C. Electronic Structure of FCC Iron The band structures of FCC iron at a = 6.5516 and 7.0 a.u. are shown in Figs. 7 through 10. Tables V and VI contain representative band widths and exchange splittings which describe variations of the bands with atomic volume. The valence band energies, at high symmetry points, can be found in Appendix B for the above lattice constants and a = 6.8107 a.u. Drastic changes in the FCC band structure occur at the L point. In the majority spin sub-band t _3 and L2‘ are quasi degenerate ^ for a = 6.8107 a.u. (Rs = 2.6616); for smaller Rg values, L2' lies below L-j and the branchings are as in Fig. 7 (a = 6.5516 a.u.) while for larger ones L2‘ lies above L 3 and the branchings are as in Fig. 9 (a = 7.0 a.u .). A somewhat similar branch switching is described above for BCC iron. Figure 7. Energy bands in ferromagnetic FCC iron along some lines of high symmetry for the majority or up spin. Energies are in (tydberg(Ry). The lattice constant is 6.5516 a.u. ENERGY (RY) Figure - - - - 0.6 0.4 0.8 0.2 . -B 0.0 0.2 r r 8 . Energy bands in ferromagnetic FCC iron along some lines some lines along FCC iron ferromagnetic in bands Energy . of high symmetry for the minority spin. Energies are are Energies spin. minority the symmetry for high of n ybr(y. h ltie osat s .56 a.u 6.5516 is constant lattice The Rydberg(Ry). in A ZQA 2 T XZWQLA K SX 44 ENERGY (RY) lines some along FCC iron ferromagnetic in bands Energy 9. Figure - - - 0.6 0.0 0.2 0 0.2 . 5 ^ 4 f ih ymty o te aoiy r p pn Energies spin. up or majority the symmetry for high of r i RdegR) Te atc cntn i 70 a.u. 7.0 is constant lattice The Rydberg(Ry). in are K SX 22 45 ENERGY(RY) Figure 10. Energy bands in ferromagnetic FCC iron along some lines some lines along FCC iron ferromagnetic in bands Energy 10. Figure - - - 0.6 0.0 0.4 0.8 0.2 r r of high symmetry for the minority spin. Energies are are Energies spin. minority the symmetry for high of n ybr(y. h ltie osat s . a.u 7.0 is constant lattice The Rydberg(Ry). in A XZWQLA XZWQLA r 25' Z KS X 46 Table V. Representative band widths (W) and exchange splittings ^ Eex> f°r ECC iron - in Rydbergs (Ry) - for various atomic radii (R3) or lattice constants (a) in a.u. Up or down spin is indicated by t or respectively. a w xt w x+ Rs ; H sp* d d Wd|tn 4Ee*(X3> 4Ee x Rydbergs (Ry) - for various atomic radii (Rs) or lattice constants (a) in a.u.. Wdcc is the occupied d band width. Up or down spin is indicated by + or +, respectively. ------r i (L,) (L,) R W°CC SE 3 <5E 3 S 1 a d d d ex ex i 2.1885 , 5.6 0.6734 0.3353 0.8799 0.0246 0.0415 2.3448 1 6.0 0.5205 0.6283 0.6749 0.0426 0.0596 | 2.5603 i 6.5516 0.3828 0.4305 0.4776 0.0782 0.0952 2.6616 , 6.8107 0.3361 0.3623 0.4109 0.1047 0.1218 2.7356 1 7.0 0.3376 0.3146 0.3887 0.1639 0.1887 | 3.5172 i 9.0 0.2620 0.0636 0.1001 0.2237 0.2310 i 49 In both cases the representation reordering occurs around Rg = 2.6616 a.u. and is limited to the majority sub-bands. For lattice parameters less than 6.5516 a.u. the exchange splittings of the two uppermost d bands (A-j branches included), almost constant in the zone, double for 0.5 a.u. increase while for values above 7.0 a.u. the splittings change by less than 10% for such increase. At a = 6.5515 a.u. the ratio of the splitting to the width for these bands are around 1/3 while their values are slightly less or greater than one for the lower and upper (as at W) ones at a = 7.0 a.u. If one limits consideration to the W-L- r branches of thes bands one still finds the ratio less than one for both bands at a - 6.5516 a.u. while it is greater than one for both at a = 7.0 a.u. The above points are particularly meaningful for the up spin sub-bands. The two uppermost d bands for the minority spin are either empty or clearly participate to the Fermi surface. However, for lattice spacings equal to or greater than 7.0 a.u. these bands are fully occupied for the majority spin. The above discussion, the first of its kind on the details of the band structures of iron will be of importance in the understanding of the magnetic properties of this metal. Some by products of these band structure calculations, using BNKPKG, are the charge and spin form factors. We shall discuss these in the next section. D. BCC and FCC Iron Form Factors The x-rays and neutron scattering form factors are gauge of the quality of the wave functions resulting from calculations as the present. This is so because these quantities, also respectively 50 referred to as charge (fc) and spin (fg) form factors, are directly related to the Fourier transform of the charge density: fc(k, 4,m) = p{k,£,m) • £2 and p+(M ,m ) - p_(k,£,m) fs(k, t,m) = p+(o,0,0) - pJ0,0,0) (32) In these expressions £1 is the cell volume and p(k,4,m) stands for the Fourier transforms of the charge density with k, fc, m designating the x, y, and z components of the reciprocal lattice vectors in units of 2ir/a. Charge and spin form factors of BCC iron are in Tables VII through IX for a = 5.0, 5.4057, and 6.0 while those of FCC iron are in Tables X through XII for a = 6.5516, 6.8107. 7.0 a.u. It is to be recalled that Van Laar and coworkers ^4 have established that the spin form factors for BCC iron for a = 5.4057, which are the same as those provided by Callaway and Wang^, agree with experiment. The content of Table X-XII constitutes the first, at the author's knowledge, published form factors for FCC iron. These data are provided for three different lattice parameters in order to allow much better comparison with experiment. Indeed, several tests have revealed that over a range of 1 a.u., the form factors can be fitted to a polynomial in a (lattice constant) of degree two for a given it vector: s fc(k, *,m) = aQ>k2m + aljk£m ■ a + a ^ kJlni • a 51 Table VII. The charge (f t ) and spin (f ) 5 form factors for BCC iron. The coordinates k, l , m of the reciprocal lattice vectors 2tt are in units of -----. a = 5.0 a.u. d k I m fc fs 1 1 0 17.6058 0.5945 2 0 0 14.3124 0.3749 2 1 1 12.2635 0.2099 2 2 0 10.8240 0.1323 3 1 0 9.7607 0.1168 2 2 2 9.0183 0.0286 3 2 1 8.4298 0.0253 4 0 0 7.9512 0.0673 3 3 0 7.6130 -0.0014 4 1 1 7.5985 0.0323 4 4 2 5.9619 -0.032b 6 0 0 5.9453 0.0188 5 3 2 5.8448 -0.0240 6 1 1 5.8341 0.0093 6 2 0 5.7271 0.0024 5 4 1 5.6299 -0.0227 6 2 2 5.5236 -0.0080 6 3 1 5.4255 -0.0102 52 Table VIII. The charge (fc) and spin (fs) form factors for BCC iron. The coordinates k, i, m of the reciprocal lattice vectors 2ti are in units of . a - 5.4057 a.u. Q k I m fc fs 1 1 0 18.2942 0.6423 2 0 0 15.1038 0.4182 2 1 1 13.0747 0.2590 2 2 0 11.5991 0.1740 3 1 0 10.4721 0.1410 2 2 2 9.6713 0.0657 3 2 1 9.0192 0.0512 4 0 0 8.4778 0.0712 3 3 0 8.1023 0.0170 4 1 1 8.0823 0.0405 4 4 2 6.2954 -0.0272 6 0 0 6.2681 0.0109 5 3 2 6.1759 -0.0216 6 1 1 6.1582 0.0033 6 2 0 6.0536 -0.0023 5 4 1 5.9642 -0.0215 6 2 2 5.8582 -0.0107 6 3 1 5.7650 -0.0125 53 le The charge (fc) and spin (fs) form factors for BCC iron. The coordinates k, £, m of the reciprocal lattice vectors 2tt are in units of ——. a = 6.00 a.u. k 1 1 0 19.1808 0.6919 2 0 0 16.1511 0.4717 2 1 1 14.1687 0.3213 2 2 0 12.6756 0.2302 3 1 0 11.4969 0.1800 2 2 2 10.6255 0.1159 3 2 1 9.9013 0.0897 4 0 0 9.2905 0.0868 3 3 0 8.8507 0.0461 4 1 1 8.8306 0.0592 4 4 2 6.7604 -0.0170 6 0 0 6.7285 0.0060 5 3 2 6.6311 -0.0150 6 1 1 6.6103 0.0001 6 2 0 6.5000 -0.0044 5 4 1 6.4091 -0.0171 6 2 2 6.2996 -0.0112 6 3 1 6.2064 -0.0129 54 Table X. The charye (fc) and spin (fg) form factors for FCC iron, The coordinates k, m of the reciprocal lattice vectors are in units of . a = 6.5516 a.u. a k I m fc 1 1 1 18.1917 0.6716 2 0 0 16.9126 0.5960 2 2 0 13.5527 0.3124 3 1 1 11.9193 0.2256 2 2 2 11.4913 0.1612 4 0 0 10.0970 0.1656 3 3 1 9.3336 0.0610 4 2 0 9.1251 0.0804 4 2 2 8.4287 0.0273 3 3 3 8.0233 -0.0096 5 1 1 8.0327 0.0585 4 4 0 7.5032 -0.0053 5 3 1 7.2574 -0.0027 4 4 2 7.1794 -0.0255 6 0 0 7.1937 0.0474 6 2 0 6.9230 0.0181 5 3 3 6.7417 -0.0317 6 2 2 6.6944 -0.0025 Table XI. The charge (fc) and spin (fg) form factors for FCC iron. The coordinates k, ft, m of the reciprocal lattice vectors 2tt are in units of a - 6.8107 a.u. d k ft m fc fs 1 1 1 18.5324 0.6684 2 0 0 17.2753 0.5920 2 2 0 13.9609 0.3243 3 1 1 12.3174 0.2347 2 2 2 11.8915 0.1795 4 0 0 10.4475 0.1646 3 3 1 9.6683 0.0752 4 2 0 9.4442 0.0876 4 2 2 8.7158 0.0386 3 3 3 8.2907 0.0056 5 1 1 8.2867 0.0586 4 4 0 7.7344 0.0038 5 3 1 7.4712 0.0037 4 4 2 7.3928 -0.0150 6 0 0 7.3925 0.0425 6 2 0 7.1112 0.0174 5 3 3 6.9302 -0.0232 6 2 2 6.8745 -0.0004 b6 Table XII. The charge (fc) and spin (fg) form factors for FCC iron. The coordinates k, I, m of the reciprocal lattice vectors 2tt are in units of -----. a = 7.0 a.u. a k SL m f c fs 1 1 1 18.7783 0.6665 2 0 0 17.5382 0.5834 2 2 0 14.2619 0.3330 3 1 1 12.6055 0.2373 2 2 2 12.1850 0.2007 4 0 0 10.6927 0.1482 3 3 1 9.9190 0.0910 4 2 0 9.6779 0.0891 4 2 2 8.9314 0.0509 3 3 3 8.4967 0.0281 5 1 1 8.4676 0.0473 4 4 0 7.9093 0.0146 5 3 1 7.6306 0.0092 4 4 2 7.5567 0.0010 6 0 0 7.5278 0.0217 6 2 0 7.2437 0.0076 5 3 3 7.0740 -0.0099 6 2 2 7.0042 -0.0024 The relevant coefficients, a*r or a^ are provided in Appendix B for the BCC iron form factors in Tables VII through IX. It is easily verified that the contents of these tables are exactly reproduced by these fits. Intermediate values, i.e., the form factors at a = 5.2, are obtained well beyond the second decimal place. Clearly, this signifies that experimental form factors can be judiciously compared to the theoretical ones for a lattice parameter equal to that existing in the conditions of measurements ( i.e ., non-zero temperature). The above locally quadratic dependence of the form factors of BCC or FCC iron is not unique to this metal. It applies equally well to lithium and copper. CHAPTER 5 MAGNETISM IN IRON The present calculations, with ferromagnetic inputs, resulted in magnetic moments, for BCC and FCC iron which are plotted in Fig. 11 versus Rs. Table XIII displays the moments, the density of state at the Fermi level as well as the characteristic exchange splittings. The latters are the energy differences between the absolute maxima in the majority and minority spin density of state curves. Density of states curves for FCC iron at a = 6.5516, 6.8107, and 7.0 a.u. are respectively shown in Figures 12, 13, and 14. BNDPKG yields from a ferromagnetic input a paramagnetic ordering provided the latter is the most stable one. In light of this, Fig. 11 indicates that BCC and FCC iron are not paramagnetic for the R$ values in the range considered. These calculations predict ferromagnetism for BCC and FCC iron but they do not exclude anti ferromagnetism^ (in FCC iron) which is not addressed here. The first section of this chapter is devoted to the discussion of the results in Table XIII and Fig. 11. The following sections, B and C address the relevance of our findings to properties of some iron based alloys after a comparison with experiment. A. Ferromagnetism in BCC and FCC Iron Figure 11 indicates a rather smooth variation of the magnetic moment per atom in BCC iron for R$ values greater or equal to 58 MAGNETIC MOMENT ifiB) versus Bohr magneton, moment ,in The magnetic 11. Figure 4.0 2.4 0.8 3.2 0.0 2.0 Wigner-Seitz radius(Rs ),in atomic units (a.u), for for (a.u), units atomic ),in radius(Rs Wigner-Seitz C(ahd uv) n FCfl cre iron. curve) FCC(full and BCC(dashed curve) . 3.0 2.5 . 4.5 3.5 4.0 5.0 59 Table XIII. Density of states (states/atom-Rydberg) for oajority (+) and minority (*) spin electron at the Fermi energy ( ( Itydberg-atoa) ^.m agnetic moment per atom (in Bohr magneton) m, and the ch a ra cte ristic exchange s p littin g delta in (tydberg (Hy) fo r BCC and FCC iron. BCC FCC m N t(EF) m 6 Rs WW Ac RS c 2.462 8.321 6.736 1.735 0.142 2.112 5.219 5.206 0.126 0.017 2.560 9.263 3.392 1.987 0.162 2.188 5.985 5.832 0.186 0.042 2.661 11.036 3.341 2.153 0.172 2.345 7.639 9.140 0.397 0.057 2.757 13.906 4.802 2.317 0.177 2.560 12.068 10.097 0.966 0.090 2.954 2.582 19.649 2.717 0.197 2.661 23.B26 10.330 1.517 0.120 3.447 2.923 60.269 3.269 0.232 2.736 2.470 13.346 2.549 0.187 4.431 1.106 484.124 3.943 0.277 3.517 2.739 36.140 3.359 0.215 Figure 12. Total density of states N(E),in states/atom per Rydberg, and the valence electron number for ferromagnetic FCC iron. The lattice constant is a = 6.5516 a.u. 42 36 S 30 o UJ 0.8 - 0.6 0.4 -0.2 0.0 0.2 0.4 0.6 ENERGY (RY) iue 3 Ttl est o sae NE, n ttsao pr Itydberg, per states/atom in N(E), states of density Total 13. Figure N(E) (STATES/ATOM-RY) 40 - 40 48 24 56 . -. -. -. 0002 . 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.8 n te aec eeto nme fr ermgei FCC ferromagnetic number for electron valence the and rn Te atc cntn i a 680 a.u. 6.8107 = a is constant lattice .The iron ENERGY (RY)ENERGY 0 ELECTRONS iue 4 Ttl est o sae NE,n ttsao pr Rydberg, per states/atom N(E),in states of density Total 14. Figure N(E) (STATES/ATOMS-RY) 44 52 36 20 . -. -. -. 00 0.2 0.0 -0.2 -0.4 -0.6 0.8 n te aec eeto nme fr ermgei FCC ferromagnetic number for electron valence the and rn Te atc cntn i a 70 a.u 7.0 = a is constant lattice The iron. ENERGY (RY) 0.4 0.6 9 2 9 63 64 2.462 a.u. This behavior of the moment is explained, in general terms, by the gradual and continuously filling of the majority spin bands as compared to the minority ones. Due in part to the increase of the exchange effect, the majority spin bands are lowered with respect to the Fermi level while the minority energy levels are pushed upwards. BCC iron becomes a strong ferromagnet between the lattice constants of 5.6 and 6.0 a.u. This corresponds to the sinking below Ep of the Gg branch in the up spin band. The smooth increase of the moment is basically the result of the progressive (see Fig. 4) transfer of electrons from the minority spin bands. A drastic change is not observed simply because no flat band crosses the Fermi level. We recall that the relatively flat bands in the structure, at ordinary lattice parameters, are the two uppermost d bands (D^, Dg, A 3, and branches) which, as previously stated are respectively full or empty for the majority or minority sub bands. This behavior of the bands, which fully explain the moment versus Rs curve of BCC iron is not unique to this metal. Walmsley and coworker^® recently have grown a BCC cobalt film, for the first time, on a chromium substrate. The moment was believed to be around that of the ordinarily stable HCP phase of this metal. Bagayoko, et al.^ performed ab-initio band structure calculations on this novel phase of cobalt at lattice constants of 5.0, 5.2346 (experimental value), 5.3099 and 5.4456 a.u. The magnetic moments these authors reported, as shown in Table XIV, behave like those of BCC iron when the atomic volume changes. The flat bands which are believed to play a stabilizing role as to the variation of the moment are equally below and above the Fermi level, for up and down spin respectively, for the lattice constants considered. Like the case of BCC iron, no drastic 65 TABLE XIV. The magnetic moment(m),in Bohr magneton, for BCC cobalt at various atomic radii( ) or lattice constants(a) in atomic units(a.u). R (a.u.) a (a.u.) s 2.4619 5.0 1.545 2.5774 5.2346 1.646 2.6144 5.3099 1.678 2.6813 5.4456 1.736 66 change with lattice parameter is expected for values of the latters above 5.0 a.u. This behavior of the magnetic moment of BCC iron is quite similar to that of FCC iron at lattice parameters above 7.0 a.u. Figure 11 shows the two curves become indistinguishable between the Rg values of 3.25 and 3.75 a.u. However, the magnetic moment of FCC iron for Rs values below 2.73 (a less than 7.0 a.u.) behave quite differently than does the BCC moment. There is a rather steep rise of the moment around Rs = 2.71 a.u. This has been first observed theoretically by Madsen and 9 Q Andersen ; these authors found a different Rg value for the transition (Rs H 2.61 a.u.). This transition has been very recently explained in terms of the band structures by Bagayoko and Callaway^5. The following explanation parallels the one given by these authors. A relatively flat portion of the uppermost up spin d band, see Fig. 7, crosses the Fermi level for a = 6.5516 a.u. Specifically we are referring to the Zg» uppermost Qj and Aj branches. These branches, as shown in Fig. 8 , are above the Fermi level in the minority spin bands for the same lattice constant. The combination of the increase of the exchange effect and the decrease of kinetic energy lower these branches, in the majority sub-bands, while it pushes them upwards in the down spin bands, relative to the Fermi level as the lattice parameter increases. For the majority spin these branches are below Ep at a = 7.0 a.u. The abrupt increase in the magnetic moment of FCC iron when the lattice constant change from 6.5516 to 7.0 is due to this transition, i.e., the passage of the uppermost up spin d bands below the Fermi energy and the subsequent disappearance of portions of the Fermi surface. This transition is continuous in the sense that these bands progressively 67 sink below the Fermi level. There is a second aspect of this transition equally as important as the filling of these up spin bands, i.e ., the increased flatness of the bands which is the manisfestation of a relative localization. We have observed, see Figs. 7 through 10, that the ratio of the exchange splitting to the width of the uppermost d bands in FCC iron is less that unity for lattice constants less than or equal to 6.5516 a.u. This ratio is a quantitative measure of the degree of flatness of these bands. For Rg values above 2.73 a.u., the ratio is greater than one, indicating relatively flat bands or an increase in atomic character of the electrons occupying them. As explained in the discussion of the magnetic properties of BCC iron, once these flattened d bands lie below Ep, in the majority spin, no drastic variation of the moment with lattice parameter is expected, because the corresponding ones in the minority spin bands are above the Fermi energy. Bagayoko and Callaway estimated the transition atomic radius to be Rs = 2,71 a.u. At the transition atomic volume Z 2 branch is expected to lie at the fermi level in the up spin sub-band. The data in Table XIII show the onset of the transition to strong ferromagnetism for BCC and FCC iron. In both cases it is indicated by a relatively large density of down spin states as compared to that for the up spin ones at the Fermi level. Figures 12, 13 and 14 illustrate the same transition for FCC iron. It should be noted that while these figures are indicative of the establishment of strong ferromagnetism, one needs the detail band structure to ascertain the otherwise conjectural observation. If the moment of BCC cobalt behaves like that of BCC iron for 68 varying atomic volume, the situation in nickel seems to resemble that of FCC iron. Indeed, Wang and Callaway^4 presented the band structure of this 3d metal for a lattice constant of 6.644 a.u. Weling and 78 Callaway produced semi-empirical bands of FCC nickel which fitted the experimentally observed7^ structure. Bagayoko7^, i n the course of testing the new version of BNKPKG obtained results qualitatively similar to those of Wang and Callaway54. The magnetic moment obtained by the various calculations is around 0.55 Ug. The important feature of the structures of FCC nickel of particular interest here is the itinerant character they possess as compared to BCC iron. The ratio of the exchange splitting to the total individual widths for the uppermost d bands of nickel (a = 6.644 a.u.) is less than one; a result similar to that found for FCC iron at a = 6,5516 a.u. However, if one considers only the r-L-W branches of these bands, in calculating the widths, the above ratio remains less than 1 for the down spin cases and becomes greater than one for the uppermost up spin (r i 2+“L3+"Wl+'^ in case of nickel. B. Comparison with Experiment Our prediction of ferromagnetism for FCC iron is in agreement with the most recent experimental conclusions 17*^5 as we shall discuss. At first a comparison with the theoretical works seems appropriate. 01 Andersen and coworkers reported two different results on the magnetic ordering in FCC iron. Madsen and Andersen^ found FCC iron to on be ferromagnetic for Rs values above 2.47 while Poulsen et a l. found a non-zero moment only for Rs larger than 2.6 a.u. Poulsen et a l., unlike Madsen and Andersen, included the sp-d hybridization in all their 69 calculations. The zero moment result of Poulsen et a l. could be attributed to an overestimate of the correction to the unhybridized results. The drastic changes in the d band branchings as described here are indeed difficult to account for by the positioning and scaling of the projected density of states as was done by Madsen and Andersen or Poulsen et al. when the sp-d hybridization is included. The rather good results these authors obtained for normal volume (i.e., a magnetic moment of 2.17 u^) do not contradict the above contention as a full band structure calculation was used in that case instead of scaling procedures. Bagayoko and Callaway^ recently discussed the results of OO Kubler . There is a qualitative agreement between this author's results and ours on the ferromagnetism of FCC iron below Rs = 2.60. There exist quantitative differences. In particular we find the moment not to vanish at Rs = 2.50 a.u. and we place the abrupt increase of the moment of FCC iron around Rs = 2.71 instead of about 2.645 a.u. The total energy curves obtained by Kubler indicate the stable phase of this metal to be an anti-ferromagnetic FCC, contrary to experiment, for Rs values between 2.5 and 2.7 a.u. The spherical averaging of the charge 32 density inherent to the augmented spherical wave method is suspected to cause this problem. Roy and P e ttifo r^ found the Tauer and Weiss y^ and Yg states to be both ferromagnetic. Their unhybridized ASA calculations are therefore in qualitative agreement with our results. However we should emphasize the continuity in the steep rise of the moment between Rs = 2.56 and Rg = 2.736 a.u. which is apparent from Fig. 1 as opposed to the discontinuity implied by the original y^ (antiferromagnetic) and Yg 70 {ferromagnetic) hypothesis. Even though it is difficult to draw a conclusion from Kubler's results, several experimental ones found antiferromagnetic FCC iron at some Rg value less than 2.76. As discussed in the first chapter, the magnetic ordering in FCC iron is suspected to depend on both film orientation2^ and atomic separation2^. The difficulty connected with the experimental determination of the lattice constant compounds the problem. Still the results of Wright seem to be in good agreement with ours. This author found a ferromagnetic y Fe film to possess a moment of 1.1 p^, for a lattice spacing estimated to be 6.8 a.u. At a lattice constant of 6,8107 a.u. the present result shows a moment of 1.517 p^. Considering that 6.8 a.u. was an upper limit for the experimental value and the rapid variation of the moment around this lattice parameter, this appears as a good agreement between experiment and theory. Gonser et a l,25 estimated a transition from anti-ferromagnetism to ferromagnetism to occur between Rg = 2.67 and Rg = 2.78 a.u. These authors, as mentioned in the introduction, found FCC iron precipitate in Cu6g Augg to be ferromagnetic. We found the atomic radius for the transition to strong ferromagnetism to be about 2.71 a.u. for FCC iron. Their estimate of the smallest Rs value (2.67) compatible with ferromagnetism in FCC iron seems to be in disagreement with Wright's result where ferromagnetism is found at an Rs value below 2.66 a.u.The difficulties in the evaluation of the lattice parameter, due in part to possible pseudomorphic phase for the film, could account for the difference and vindicate either finding. C. Relevance to Iron Based Alloys 71 The present study of ferromagnetic y iron is quite relevant to the properties of FCC iron based alloys in general and the invar alloys in particular. That is so in part because at most two shells of neighboring iron atoms are believed to be needed for the central atom to op exhibit bulk properties . The most recent explanations of the invar anomalies do not assume y iron to be antiferromagnetic^,®^,®*»®^. While the explanation of pi Q p Kaspar and Salahub ^1 assume ferromagnetic order, Gavoille reaches the conclusion that antiferromagnetically coupled iron atoms cannot be responsible for invar phenomena. The band structure of FCC iron in Fig. 7 through 10 could be used to support arguments of the type presented by Kaspar and Salahub. These authors obtained the ferromagnetic energy levels of Fe^, Fe^Ni and Ni^ clusters; the respective non-bonding and anti-bonding characters of the minority and majority spin sub-levels just above the Fermi level (Ep) are used to discuss invar effects in ^e64 N^36 concentrations around 35%) through filling and emptying of levels. Two factors entering the analysis are the smaller atomic volume and excess electrons ( 2) nickel possesses as compared to OO iron. Hattox and coworkers studies some properties of BCC vanadium as a function of the lattice constant. They described the possible usefulness as well as the limitations of the properties of pure metals in understanding alloys. The relevance of the present results to a complete theory of invar phenomena is illustrated by the work of Shiga 8 4 who studied the magneto-volume invar effects for Fe-Ni alloys among others. This author fits rather well the lattice constant (a(X)) versus nickel concentration (X) curves of Fe-Ni alloys with a modified Vegard's Law: 72 a(X) = ax (1 - X) + a 2 X + c y(X) where a^, a.? and c are adjustable parameters and y(X) an average atomic moment. At absolute zero the curve shows the maximum lattice constant for the alloy to be about 6.77 a.u. at approximately 38% nickel; this spacing is in the region of steep rise of the moment of FCC iron. The deviations from the Vegard's Law are attributed to a magnetic £ contribution to the expansion as recently discussed by Morruzi et a l. for pure metals. CHAPTER 6 DISCU SSIO N AND SUMMARY It is apparent from the survey of previous works as well as the precediny description of magnetism in BCC and FCC iron that the present results and their analysis could be of importance for future investigations. We shall elaborate on such investigations as they appear through further discussion in section A. We present a summary of our findings in section B. A. Discussion for Future Work While experiments agree on the absence of paramagnetism in FCC iron, as found here, they disagree somewhat on the magnetic ordering in this metal. Anti ferromagnetism ^ * ^ 4 and ferromagnetism^ *22,23,25 are OA found in both gamma iron precipitates and films. Film orientation or pc atomic separation ^3 are suggested to be factors determining the magnetic order in FCC iron. However, we mentioned the observation of ferromagnetism in iron film ^ at a lattice constant smaller than the O |T minimum required, according to Gonser et a l., for the establishment of this magnetic ordering. The present results could allow meaningful comparison with experimental ones provided progress is made in the determination of the atomic separation in films or precipitates. A first possible continuation of this work would be to extend the band package, BNDPKG, in order to produce the band structure of materials with two atoms per cell. The treatment of anti ferromagnetism requires such a modification. These changes could be made in a manner 73 74 to allow for the calculation of the total energy. Although it might still be difficult to accommodate such factors as pseudomorphism in films or defects, this will permit the determination, from the total energy curve, of the most stable ordering at a given atomic volume for the pure metal. This is needed because any inference of the stability 32 of the anti ferromagnetic ordering from Kubler's results requires iron to be in the FCC structure, contrary to reality, in ordinary conditions of temperature and pressure. These total energy calculations could be supplemented by the concomitant evaluation of the pressure. Indeed the infinite terms which appear in the expression of the pressure as given by Janak®^ are easily avoided by expressing the Coulomb contributions to the potential energy in terms of U and D as defined in Chapter 2. This of importance as previous evaluation of the pressure necessitated approximations, like the Muffin-Tin one,®*’ in order to avoid divergences. All the required entities are obtained in the process of band structure calculations using the new version of BNDPKG as shown in appendix c. The above total energy and pressure calculations might require an large amount of CPU time for iron due to a slow convergence of the sums in U and D as defined in Chapter 2. This is so because of the very large gaussian exponents used to describe the core and also the nuclear repulsion term in the potential. In the work on copper the sums had to be carried out to a value of the square of Kg over 12,000 to obtain convergence. Such a summation appears impractical at over 20 lattice parameters as would be the case here for iron. Based on the rather slow variations of the Fourier coefficients at large Kg values one could replace the sums by integrals. This non-trivial honing of the total 75 energy calculation procedure is not only of potential use in the above suggested work on iron but also will enable one to study, from band calculations, the central pair potentials as described by Matthai et_ ftfi ft7 a l. and March . Besides their obvious usefulness in metallurgy, these pair potentials can by applied to the study of defects.8^ An important result obtained here has been the observation of the atomic behavior of some valence states in BCC iron. In FCC iron we have similar results for lattice constants above 7.0 a.u. These characteristics show up in the values and variations of the band widths QQ QQ and exchange splittings. Experimental and theoretical works OQ intimated these results. In particular, Stearns obtained evidence for quasi-localized states in BCC iron as well as other 3d metals which are hybridized with itinerant or continuum ones. The shake up structures 7Q QQ observed in nickel could be seen as further confirmation of Stearn s observations. The increased directionality 86 from FCC to HCP and BCC, in that order, implied a somewhat atomic character in BCC iron. A more direct observation of this fact was the failure of the cluster approach calculation to yield an acceptable value of the critical temperature (Tc) of iron for any size of the cluster. On the contrary, Ziegler 89 reproduced the experimental value of this quantity for nickel. Despite the above wealth of information on the localization of valence electrons in BCC iron, a detailed conceptual understanding of it is not yet available. It is not clear what interpretation one can give to parabola like portions of the bands they occupy. Various forms of hybridization (sp-d, d-d) can be invoked but lack a firm footing. The data obtained here can be used to compute the probability that an electron occupying a band no at a given site at time t be found at 76- titne t 1 at another site. The basic formalism to do this has been provided by Callaway and Hughes 90 in their study of localized defectsin semi-conductors using Wannier functions. Such transition time or probability will allow a quantitative description of the degree of localization at any lattice constant. The above suggested experimental as well as theoretical works can benefit from our findings. In particular most computations we mentioned can use the present data as input. Other studies as those of the Compton profile, the Fermi surface, or the optical conductivity can, in a straight forward manner, utilize the results we wish to summarize in the following section. B. Summary We have reviewed previous theoretical and experimental works on BCC and FCC iron. A new formalism for the calculation of the total energy and pressure of a metal has been described. We extended Wang and Callaway's program package in order to implement this formalism. We described the general and new features of this package, BNDPKG, which utilizes Gaussian basis in a LCAO scheme. Local density potentials (i.e ., VBH, RSK) are allowed for in BNDPKG. The large size of the Hamiltonian matrix resulting from the use of GTO or CGTO led to a careful study of contraction of basis sets. The most important successful test of the guidelines set forth for contraction has been the obtention of a minimum, at about the experimental lattice constant, in the total energy curve of FCC copper, a 3d metal. The large reduction of CPU time resulting from the use of CGTO opened the way for the present detailed study of the electronic structure of BCC and FCC iron. 77 Keeping in sight possible extentions of this work, as suggested above ( i.e ., total energy and pressure calculation), we employed the new version of BNDPKG. The band widths and exchange splittings discussed quantitatively described the variations of the electronic states in BCC and FCC iron as the lattice constant changes. The decrease or increase of the band width or the exchange splitting as the lattice parameter increases is explained by the competition between the kinetic and exchange energies. Several features of the electronic structures of iron have been reported here for the first time. In particular the reordering of representations in the band structure of BCC and FCC iron was described. This reordering, which occurs for both structures around Rs = 2.6616 a.u. is accompanied by changes of branching of the band to avoid forbidden crossings.This branch switching is suspected of being a contributing factor to the difference between the results of Madsen and Andersen*^ and Poulsen et al While ferromagnetism in BCC iron was found to possess an atomic origin due to characteristically flat portions of the d bands, the results for FCC iron are mixed. We explained, using ab-initio self- consistent band structures, the abrupt jump in the magnetic moment of FCC iron when Rs varies from 6.5516 to 7.0 a.u. This jump is found to be the result of the establishment of strong ferromagnetism in FCC iron. We emphasized the increased flatness of the top d bands in FCC iron for Rg values above the transition point (Rs = 2.71 a.u.). For very large lattice constants we found the moment in both structures to behave similarly. We found in BCC iron, at lattice constants above 9.0 a.u., the crystal field splitting to be larger than the overlap effects. The forms factors we obtained as well as their locally 78 quadratic dependence on the lattice constant could be of use in the analysis of experimental works. BCC cobalt has been predicted to behave, as far as the moment variation with Rs is concerned, like iron. The reasons for this were given in terms of the band structure. We indicated the usefulness of our results in a complete understanding of iron alloys in general and the invar alloys in particular. We mentioned few of the many possible future works, experimental or theoretical, for which the results of this work, presented or available,9* could play a role of reference material. REFERENCES 1. R. A. Tawil and J. Callaway, Phys. Rev. B7_, 4242 (1973). 2. J. Callaway and C. S. Wang, Phys. Rev. B16, 2095 (1977). 3. J. Callaway, Institute of Physics Conf. Ser. 55, 1 (1981). 4. E. P. Wohlfarth, Ferromagnetic Materials, Vol. 1, 1 (1980). Edited by E. P. Wohlfarth, North-Holland Publishing Co., Inc. 5. D. E. Eastman, J. F. Janak, A. R. Williams, R. V. Coleman, and G. Wendin, 0. Appl. Phys. 50, 7426 (1979). 6 . V. L. Moruzzi, J. F. Janak, and A. R. Williams, Calculated Electronic Properties of Metals, (1978). Pergamon Press, Inc. 7. H. Danan, A. Herr, and A. J. P. Meyer, J. Appl. Phys. 669 (1968). 8 . P. Heiman and H. Neddermeyer, Phys. Rev. B18, 3537 (1978). 9. A. M. Turner and J. L. Erskine, Phys. Rev. B25, 1983 (1982). 10. I. V. Svechkarev and A. S. Panfilov, Phys. Stat. Sol. (B)63, 11 (1974). 11. L. Kaufman, E. V. Clougherty, and R. J. Weiss, Acta Metallurgica, JJL, 323 (1963). 12. L. Vinokurova, Soviet Phys. JETP 49(5), 834 (1979). 13. L. Vinokurova, and E. Kulatov, J. MMM 15-18, 1205 (1980). 14. L. G. Liu and W. A. Bassett, J. Geoph. Res. jW, 3777 (1975). 15. H. M. Strong, R. E. Tuft, and R. E. Hanneman, Met. Trans. 4_, 2657 (1973). 16. F. P. Bundy, J. Appl. Phys. 616 (1965). 79 80 17. P. J. Brown, H. Capellman, J. Deportes, D. Givord, and K. R. A. Ziebeck, J. MMM 31-34, 295 (1983). 18. W. B. Pearson, Handbook of Lattice Spacings, Pergamon Press, London (1959). 19. K. J. Tauer and R. J. Weiss, Bull. Amer. Phys. Soc. 6_, 125 (1961). 20. R. J. Weiss, Proc. Phys. Soc 82_, 281 (1963). 21. J. Kaspar and D. R. Salahub, Phys. Rev. Lett. 47, 54 (1981). 22. J. G. Wright, Philo. Mag. _24, 217 (1971). 23. U. Gradmann, N. Kummerle, and P. Tillmanns, Thin Solid Films 34, 249 (1976). 24. W. Keune, R. Halbauer, J. Gonser, J. Lauer, and 0. L. Williamson, J. Appl. Phys. 48, 2976 (1977). 25. U. Gonser, K. Krischel, and S. Nasu, J. MMM 15-18, 1145 (1980). 26. J. H. Wood, Phys. Rev. 126_, 517 (1962). 27. R. A. Deegan, Phys. Rev. 171, 659 (1968). 28. N. W. Dalton, J. Phys. C: Solid State Phys. Vol. 3, 1912 (1970). 29. J. Madsen and 0. K. Andersen, AIP Conf. Proceed. 29, 327 (1975). 30. U. K. Poulsen, J. Kollar and 0. K. Andersen, J. Phys. F: Metal Phys. Vol. 6 , No. 9, L241 (1976). 31. 0. K. Andersen, J. Madsen, U. K. Poulsen, 0. Jepsen, and J. Kollar, Physica 86 -88 B, 249 (1977). 32. J. Kubler, Phys. Lett. 81A, 81 (1981). 33. A. R. Williams, J. Kubler and C. D. Gelatt, J r ., Phys. Rev. B19, 6094 (1979). 34. D. M. Roy and D. G. Pettifor, 0. Phys. F: Metal Phys. Vol. 7, No. 7, L183, (1977). 35. E. C. Stoner, Pro. Roy. Soc. A169, 339 (1939). 81 36. H. Hasegawa and D. G. Pettifor, Phys. Rev. Lett. 50, 130 (1983). 37. H. Hasegawa, J. Phys. Soc. Japan 4j>_, 1504 (1979) and 49^ 963 (1980). 38. J. Hubbard, Phys. Rev. _B19_, 2626 (1979) and _23_ 5974 (1981). 39. W. Kohn and L. J. Sham, Phys. Rev. A140, 1133 (1965). 40. L. J. Sham and W. Kohn, Phys. Rev. 145, 561 (1966). 41. Hohenberg and W. Kohn, Phys. Rev. B136 (1964). 42. U. von Barth and L. Hedin, J. Phys. C5, 1629 (1972). 43. J. Callaway, Quantum Theory of the Solid State, Academic Press, N.Y. (1976). 44. F. Bloch, Zeit Physik 52_, 555 (1928). 45. C. S. Wang and J. Callaway, Comput. Phys. Commun. _14_, 327 (1978). The program package CNDPKG is available from comput. phys. commun, program library. 46. A. K. Rajagopal, S. P. Singhal, and J. Kimball (unpublished), as quoted by A. K. Rajagopal, in "Advances in Chemical Physics," edited by G. I. Prigogine and S. A. Rice (Wiley, New York, 1979), Vol. 41, p. 59. 47. J. Callaway and N. H. March, Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull, Academic Press, N.Y., to be published. 48. A. R. Williams and U. von Barth, Applications of Density Functional Theory to Atoms, Molecules, and Solids. 49. J. Callaway, Xianwu Zou, and D. Bagayoko, Phys. Lett. A89, 86 (1982). 50. J. Callaway, Xianwu Zou, and D. Bagayoko, Phys. Rev. B27, 631 (1983). 82 51. C. S. Wang, J. MMM 31-34, 95 (1983). 52. 0. Gunnarson, M. Jonson, and B. I. Lunquist, Phys Rev. B20, 3136 (1979). See also 0. Gunnarson and R. 0. Jones, Phys. Sci. 21, 394 (1980). 53. S. H. Vosko, L. Wilk, and M. Nusaiy, Can. J. Phys., 1200 (1980). 54. C. S. Wang and J. Callaway, Phys. Rev. B15, 298 (1977). 55. J. P. Perdew and M. R. Norman, Phys. Rev. B26, 5445 (1982). 56. J. D. Pack, H. J. Monkhorst, and D. L. Freeman, Solid State Commun. 29, 723 (1979). 57. J. Boettcher, private communication. 58. E. E. Lafon and C. C. Lin, Phys. Rev. 152, 579 (1966). 59. S. F. Boys, Proc. Roy. Soc. (London), A200, 542 (1950). 60. I. Shavitt and M. Karplus, J. Chem. Phys. 36, 550 (1962). 61. R. C. Chaney and F. Dorman, Int. J. Quantum Chem. 8 , 465 (1974). 62. D. Bagayoko, D. G. Laurent, S. P. Singhal, and J. Callaway, Phys. Lett. 76A, 2 (1980). 63. A. J. H. Wachters, J. Chem. Phys. _52_, 1033 (1970). 64. A. Veillard, Theorit. Chim. Acta (B e rl.),^ 2 , 405 (1968). 65. E, Clementi and D. A. Davis, J. Comput. Phys. _1_, 223 (1966). 66. S. Huzinaga and Y. Sakai, J. Chem. Phys. jSO, 1371 (1969). 67. T. H. Basch, C. T. Hornback, and J. W. Moskowitz, J. Chem. Phys. 50_, 1311 (1969). 68 . R. F. Steward, J. Chem Phys. _50, 2485 (1969). 69. C. Salez and A. Veillard, Theorit. Chem. Acta (Berl.), _U_, 441 (1968). 70. W. Kutzelnigg and W. H. E. Schwarz, Phys. Rev. A26, 2361 (1982). 83 71. D. Bagayoko, Int. 0. Quant. Chem. Proceed. 1983 Sanibel Conf., in press. 72. B. Roos, A. Veillard and G. Vinot, Theorit. Chem. Acta (Berl.), ZOj 1 (1971). 73. D. Bagayoko, unpublished. Resulting bands are available upon request. 74. B. Van Laar, F. Maniawski, S, Kaprzyk, and L. Dobrzynski, J. MMM 94 (1979). 75. D. Bagayoko and J. Callaway, Phys. Rev. B, to be published. 76. R. Walmsley, J. Thompson, D. Friedman, R. M. White, and Th. H. Geballe, (at Stanford University), to be published. 77. D. Bagayoko, A. Ziegler, and J. Callaway, Phys. Rev. B27, 7046 (1983). 78. F. Weling and J. Callaway, Phys. Rev. B26, 710 (1982). 79. W. Eberhardt and E. W. Plummer, Phys. Rev. B21, 3245 (1980). 80. A. R. Williams, V. L. Moruzzi, C. D. Gelatt, J r ., and J. Kubler, J. MMM 31-34, 88 (1983). 81. 0. Yamada, F. Ono, I. Nakai, M, Maruyama, K. Ohta, and M. Suzuki, J. MMM 31-34, 105 (1983). 82. G. Gavoilie, J. MMM 13, 255 (1979). 83. T. M. Hattox, J. B. Conklin, J r., J. C. Slater, and S. B. Trickey, J. Phys. Chem. Solids 34, 1627 (1973). 84. M. Shiga, J. Phys. Soc. of Japan 22, 539 (1967). 85. J. F. Janak, Phys. Rev. B9, 3985 (1974). 86 . C. C. Matthai, P. J. Grout and N. M. March Int. J. of Quant. Chem. Symp. 12, 443 (1978). 87. N. M. March to be published. 84 88 . M. B. Stearns, AIP conf. proceed. 29, 286(1975). See also Physica 91b, 37 (1977). 89. A. Ziegler, Phys. Rev. Lett. 48, 695 (1982). 90. J. Callaway and A. J. Hughes, Phys. Rev., 156, 860 (1967). 91. Supplementary data available upon request from D. Bagayoko or J. Callaway. APPENDIX A 85 86 APPENDIX A.l: New Version o f PROGRAM FCOF DSN=FBBAGA. DIOLA.PCOF //FEFIB160 JOB (1103,72121,20,2).■PH.DIOLA*, MSG1EYEL*1, / / NOTIFT=PHEAGA / • AFTEB FEF1B060 /•JOBPABM SHIFT** /•BOOTH PHI NT PH1SICS //A EXEC FOETHCLG,PABN.FOBT=,N050UBCE,11NG1VL(66),OPT(2) ✓/ FARM,. LRED2 MOXBEF,REGION-12COK,TIME=999 //FOBI.SISIB DD • C c PFOGRAH FCOF c THE FIRST PART OF c c A GENERAL PROGRAM TO CALCULATE SELF-CONSISTAUT ENERGY c c USING THE MODIFIED TIGHT BINDING OH LCGC METHOD c i c BT c c C. S. BANG AND J. CALLANAY c EXTENSION FOB TOTAL ENEBGY CALCDLATIONS c BY c J. CALLANAY,X. ZOU.ANE D. BAGAIOKO c c DEPARTMENT OF PBYSXCS rV c LOUISIANA STATE UNIYEBSITX c c BATON BODGE LOUISIANA 70803 c c c c COMMON ELCCKS OCCUB1NG IN FCOF c c B0UT1KE5 USED IN FCOF c FOB TOTAL ENERGY CALCULATION c ATDENS ATDEN1 c CFHYGF c COULOM c FI LON FII01 c GBZPT c GIRDPK c GMIS c GPEHMK c GBADPT c OPENAD c POEXCH POEXC1 c SPHUT c TELECT c c VXCRS1 c c c c COMM. SIS, SOUTINES IN HHICH THEY API USED c c CHARGE ATDENS,FCCF,OPENAC,POEXCH c CONST ATD2NS,COULOM,FCCF,OPENAD.POEXCB c END FCOF,OPENAD 87 IN PROGHAB. THIS t DSN*PHBAGA.DIOLA.FCOF AXCU AXCU ACA FCOF,POtXCH FCOF C C C VKO ATDENS,COULOfl,FCOF,OPENAD,POEXCH C C C LCfi FCOF c c C IB FCOF USED CHANILS INPUT/OUTPUT C PUNCHER CARD FT07F001 c C (K**2.IE.EAXK2) FOB C COEFFICIENTS FOURIER FT01F001 C FT02F002 AND IN C FT01F002 DATA C SCF2. CONTAIN FT11F001 C ABC AND FT10F00J NT,SCF1 IN ESI 05ED C HEADER CAED FT05F001 FEINTER LINE C FTQ6F001 c ONES. NEEDED PIUS c ADDITIONAL GENERALLY THIS IS ROT C FT01F002 FOURIER COEFFICIENTS FOE (K**2.G1.BAXX2) IN ESINT 05ED C (K**2.G1.BAXX2) FOE C (K**2.LE.BAXK2) FOB W COEFFICIENTS OF IR COEFFICIENTS C FOURIER FT01F002 N (K«*2.GT.8AXK2) FOR OF FODIII COEFFICIENTS FT02F001 FOURIER FT02F002 C EACH ATOM IS SUEAOUNDED 61 TliO SPHERES OF EACIUS BOGAUS AND EOS PH. EOS AND BOGAUS EACIUS OF SPHERES 61 TliO C IS SUEAOUNDED ATOM EACH C C FOLLOWS: C THE FOURIER COEFFICIENTS OF THE C0ULCB3 POTENTIAL ARE EVALUATED ARE POTENTIAL C0ULCB3 NAT. THE C OF FOLLOUING C IN COEFFICIENTS THE FOURIER C THE IS EVALUATED C ANALTTICALLY. PCTENTIAL SYMMETRIC. IS EXCHANGE C SPHERICALLY THE DENSITY CHARGE TEE C IF(R.LT.ROGAUS), POINTS. C USE ABE C FUNCTIONS NAVE TXPE GAUSSIAN c POTENTIAL. EXCHANGE AND C COULCBB THE OF COEFFICIENTS FODBIEE C THE FODBIER COEFFICIENTS OF THE EXCHANGE POTENTIAL ARE OBTAINED ARE POTENTIAL EXCHANGE THE C OF COEFFICIENTS FODBIER THE C ITS FOURIER COEFFICIENTS AFE OETAINEC EY FILON’S RULE AT NRPNT AT RULE FILON’S EY C OETAINEC AFE COEFFICIENTS ITS FOURIER C EBONICS. HA C BY THE FILON'S INTEGRATION METHOD BASED ON NFL LON PCINTS BETNEEEN PCINTS LON NFL ON BASED C METHOD C INTEGRATION FILON'S THE BY SPHERES. TNO EVALUATED. C ARE COEFFICIENTS FOURIER THE C SPHERE. SEITZ C C THE EXPANSION COEFFICIENTS ARE OBTAINED FBCB CHASGE DENSITY ALONG DENSITY CHASGE FBCB OBTAINED ARE C COEFFICIENTS AS C ARE EXPANSION SPHERES THE DIRECTIONS. (DIBX,DIRY,DIRZ) IIN NEIGHBORING BETH C REGION FBCB HHEN A/(2*IBCELD). CUBE EACH CONTHBUT1CNS INSIDE OF C LENGTH THE CUBES HIGHER INTO LINEARLY C THE OF IS BY ITS DIVIDED RADIUS REGION THE THE TO UP IS IS EXTRAPOLATED C DENSITY**1/3 APPROXIMATED IS DENSITY INTEGRATED C AND CHARGE THE (IGRLV.EQ.0) IF AVERAGE C SPHERICAL C ,L*0,INFINITE) ,H*-L,L) H. L TYPE P)=((4*P1*I**L*JL(K*B)*KLB(K)*KLH(F) ORDER C OF EXF(I K. HARMONIC IS CUBIC RLB(R) THE C IF (BOGAUS.LT.B.IE.ROSPH) , TBE CHARGE DENSITY IS EXFANED IN KUBIC IS EXFANED DENSITY C , CHARGE TBE (BOGAUS.LT.B.IE.ROSPH) IF C JL(KR) IS THE SPHERICAL BESSEL FUNCTION OF ORDER L. ORDER OF C FUNCTION BESSEL JL(KR) IS SPHERICAL THE o o U C C * A * i tn 1 # CD *- 1 « H M r- o 1 * I H a a h k O n H H i O »6 H H B (/) -x. — 4 W II It 4 * b tf — in — r* — h ft H W M a O PM SBkLLEB ) M 16 * H H C o * 3 a 6 o 4 • a a m o o o — m —* > 4 a N H X 6 O iA O I H «N X -* • O u o o 4 • « ftQ O 4H 0 H U ft oa ui h x x m • 1 u %X k H H A H H M 4 a M -ft tf ■» tf A *ft X a (9 • a US *l/> ft fttf» A i H h ftO w O K B i M M 4 I o m M»- ft %a o a a O • S H O W — P* •" THE l A k PJ (kt tS) 3 © VC CD o o o o w n M H PB 3 C N V h H o v m W H H H re W K w o o n O 00 O I H M D H H ID to p ► H w w k m n w k n w S ► e K)*tWO t? 0 w 0 0 * wo O 0 0 0 B to tn S OO^i K o » w o0 a o : t - e o tv (A to ^ O o w r p K H ftB m n O n II «»H W a DMHO 0 0 H 4 Z H K H a h w w 1 to m N H «r to to ii * to a k ae ► - P 5 ^ M 0 3 WHO H H V lH n •a H H H o n ► H to H** . 0 H a 0 H H 0 I o o 0 O O M t o o a WMHsoaavito u n K) 0#*. 0 n w o o o n u i x o O O 0 M N HOHCj* O H a o M I W 0 o M H o « U 0 0 o% h ci Hi H O O M O U» ►* MOM n\ so-* ** * n n n w H H * \ z w t* •m Ul ^ rt H I II H H (V ij> in o to o if h B Ci Ci M H I to M j M M O* « ten v OHHO' I m a a c to 0 N O X 0 K w h at* g ff* H — to to H to w » ♦ H o o o tr 1 W10 K z0 4 M 0 O H -* W O H 0 0« * AC U' U H H to U w -* to n h ii » m ¥« d u t to t-1 ii 2 0 * H Ci H I H II D H 9T .i H • w o • r> H DSN*PHBAGA.DIOLA.FCOF DO) /B.DO DO) C (ID)-AX (I) *C1BZ (I D) . EQ. 0. 0. AND. 0. 0. . EQ. D) (I *C1BZ (I) (ID)-AX C j DIBY (1) =AY (2) =AY (1) DIBY DIBX (1)=AX (2) (1)=AX DIBX ND*=ND-1 ND*=ND-1 140 ID=1aNDR DO DIBI (HE) *11 (I) *11 (HE) DIBI (II) (I1)=AHALF*KKX AX AX (1) = NKX(1) (I) = Y (1) AX =KK (Z) AT 170 TO GO IF(ND1BC.EQ.1) D1 RZ (1) - AZ (2) AZ - (1) RZ D1 IS1=I+1 TO 120 GO DO 160 ND=2(ND1BC 160 ND=2(ND1BC DO 150 1=ISCL,1D1BC DO BNO4l>1.25D0*DSCBT (21. CO) (21. BNO4l>1.25D0*DSCBT 12=12+5 12=12+5 DIBX (ND) = AX(I) = (ND) DIBX (II) fcALF*KKI =A (11) AY BN061*231.D0*D5QBT(26. NF1DIE=NFIL0N* NDIBC NF1DIE=NFIL0N* DIKZ (ND) = AZ (I) AZ = (ND) DIKZ 160 TO GO (1ST.LE.1DIHC) IF 1ECTOB5 LATTICE BISECT PEHRUTED GEBEBATE ,1DIB 1 19011= DO FACT*1.DO/OBEGA CEXPA*CEXFE F(-8ETAI*H**2)/B♦ALPHA*EXP(-BEIA2*B**2)/R EX ALPHA)* V(R)=-(2*Z+ IF (AY (I) *D1RZ(IE)-AZ(I)*DIRI (IE) .EC.0.0. (IE) ABC. *D1RZ(IE)-AZ(I)*DIRI (I) (AY IF .IS 1=3 .IS ST I = CL IS HALF*A 1) A El). (IECUS. IF CNCO 1* 1.D0/D5QET (4.D0*F1) 1.D0/D5QET 1* CNCO BN 081*65. DO*DSCB1(561.EO)/16.DO DO*DSCB1(561.EO)/16.DO 081*65. BN CH061=EN061*CN001 OPENAD. IN SUEBCUTINE GENERATED ABE PABARETEBS THE IF(IESALD.EU.O) CALL GPEBBK(KKX,KKY,KKZ,KSC#IDIHf3EC,BAXR2,1,0) CALL CO 1= ELECT* PI*8.DO/CBEG A PI*8.DO/CBEG ELECT* 1= CO CNC41=BNC41*CN001 CNC41=BNC41*CN001 1*BNC81*CNC01 CNC8 CDENS*4.D0*PI/OBEGA*CNC01 CEMD*2.DO*PI/OBEGA SIXPI=-6.DO* (3.DO/ (4.DO* PI))**ON ATBD PI))**ON (4.DO* (3.DO/ SIXPI=-6.DO* CEXFE*CDENS POTENTIAL. TYPE EGALD IK THE I I 1=1BCELD+1 PAEABETEBS IN THE BEAD OB GENERATE & & *D1B (I) AZ £ £ .EQ.0.0) GOTO 150 AX(I)*DIEY(ID)-AY(1)*DIBX(ID) 130 AZ (1) = NKZ (I) NKZ = (1) 130 AZ 140 140 CONTINUE 180 180 CONTINUE 190 AZ (II) =AHALF*K KZ (II) KZ (II)=AHALF*K 190 AZ 150 CONTINUE 150 CONTINUE 160 CONTINUE 170 CONTINUE uu u u u u v u non nnnn non CONTINUE102 200 23 CONilNOE0 NB220 (I) *0 4 12=12*1240 210 5 CONTINUE250 0 FORNAT(//,2I,«IEBA1D=«,HO,*101 ALPHA=*. 115. 4, 'BETA 1=*,F15.4, FOBS AT (110 ,3 ?15. 4) CONTI NOE £0,1SORT,IKC£LD,ASPAT,BFTIN,NRPNT ,EA* F5*, (i* ,E20. 1,BETA1*,5,//) V(ii)*, ,F15,**,, =12 J J=11 11*0 (IG IF A US. NE.O) GC TO 300 BEAD (5,200) IEHALE,1LPHA,BEIA1.BETA2 IF.Z (J) =KKZ (I) CALL OPEN AD (ALPHA, BET A1, EET A2, B2,5. D-1, 2. D-1, 50) GO 250 TO 11 R Y = C O RB * X K = K C Y O ( 1 f ) i * * C K G K R X H ( 1 A )* F C * O C R C H B A F + C O B C ASP L L H N T ( I 5 K X , A N Y , K K Z , N S P N T , E H O F B F , R O S P H , C O B , S O B V , 0 1 9 E G A ) 82 = (AX (2) *AX(2) *AY(2)*AY (2)*AZ (2)*AZ (2) 0 .D )/4 IB Y(J)=KKI(1) VPT2=ALPHA*DEXP(-BETA2*£2) /DSQRT(B2) F(BI.QO GCIF (NB(I).EQ.O) TO 240 J= J* BO01FY THE WEIGHTING FACTOE FOB THE COSES THAT ARE INTERCEPTED IF(1GB1V0.EQ.Q) BY GO TO 300 X R X { J ) = K K X ( I ) GO 10 TO 2 12 =HFT1N GO TO 230 DO 1=1,NSPNT 230 J=0 COBHAF-COB/2.D0 THE SPHERE. DEFINES EQUALLY SPACED POINTS 1/4BTH CF IN THE ONIT CELL. VPTt— (2. DO*ELECT*ALPHA) *DEXP(-BETA1*B2)/DSQRT (B2) MFTIN=J IF(BX*AX*£Y*BY*BZ*BZ.LT.BOSPH*BOSPH)R Z = C O R * K K Z ( I ) * C C E H A GO F * TO C C B SELECT POINTS THAT ABE INVOLVED CALCOLATING IN THE GBADIENTS. H B 1 T E (6 , 6 0 0 ) A , B O , E L E C T , E H 1 G , I D C U B , N K P T , B 1 X K 2 ,1 D I ( ! , B A XCALL B 2 , GBZPT(KKX,RKY,KKZ,aHOBHF,NSPNT,IDC,III,SUHW,1,0,1) N S T A . I G R L V IF (1EEA1C.N£.0) GO TO HR VBIS3=VPI1*VPT2 DO 1-1,NSPNT 260 NB|I)*1 B F T I N * N 3 P N T BT (J)-RHCBBF (I) = 1TE 11*1 1 , 6 ( 101)IEHA1D,ALPHA,BE1A1,BETA2,VRTST 210 DSN=PHEAGA-D10LA.FC0F 220 92 n o n IH-2.VXCB51 OSED IF IS TO CALCULATE H(BS) C 34 CONTINUE 0 3 CONTINUE330 32 CONTINUE0 1 CONTINUE310 0 CONTINUE307 CONTINUE300 0 BEAD(5305 ,EN0s 0 ,3 610)KST,KEND,KINC«,IGBIV,KPTFBT 9 COSINE290 (I,K)*DCOS (BASK)*SIA 6 COSINE260 (1,K)*D51N (AK)/AK 7 COSINE270 1)*1. (I, DO CONTINUE260 £NSFI N,IVXCB) 6CHFTIN,IVXCB,BbCBVI) CALL FOEXC1(AX,AY,AZ,1DIR,NPILON,HFI10N,F1EHCB,NDIBC,NDIBC,0, B F - B F * E B N G S Z CALL ATDEN1(AX, AY,AZ,IDIfl,IBX.1BX,IBZ,CCR,RFXIB,BHCiflF,HT,NSPIN, RNGSIZ(I)*BF DO 1*1,NNGSIZ 340 D K N G S Z - ( R O N G S Z — B O S P H ) / D F L O A T ( N URFILON(I)*BF G S I Z - 1 ) DO 1-1,NFILON 330 DPFILN*(R0SPH-R0GAU5)/LFLOAT(NFILON-1)CALL FOEXC1 (AX ,AY,AZ,IDIR,NBPNT,B,BHCR,1 NDIBC, , ,NSPIN,IVXCB) 1 DO 3201*1,NRPNI BF-BOSPii BF-RF*DRFILN BF*NOGAJS C A L C U L A TTH ECH E A B GDE E N S I T Y . CRFTIN*1.D0/(SURN*CDENS*6.D0) R F - B F + DB(I) *RF R G A U S HF*0.D0 D B G A U S - R O G A U S / D F L O A T ( N R P N T - 1 ) SINE (I,K)*DSIN (BANK)BAKK-BA-BKBA*EA«COR *CCA DO 1*2(III 290 BA-0.DO C O ASI * (A-DSIN D C O (AKJ/AK S ( A K ) ( l l -51A)/ . D 0 * A K*2,KKDO K ) 290 BK-BK+AKR BK*BK»AKR DO 2 K*2,KK 60 A K = B K * C O R B A F BK*0.D0 A K - N K * C O B H A P BK*0.CO RK=0 CALL GBADPT(NPK,IBX,IR Y,IBZ,IPI,IDVDIB.NSEHT ,1 DC) SI HE (I,1)*0.D0 SINE (1,K)*0.D0 IDVDIfl=IlI*1 DO 1*1,111 270 K M S Q t t T ( F L O A T( M A X K 2 ) }* 2 DSN*PHBAGA.DIOLA.FCOF 93 94 DSN-PHBACA.EIOLA.FCOF GINDPK (KKX,KKT,KKZ, BSC, AKPT,II!C0B,BAXK2,MB, 1,ISOBT,0) AKPT,II!C0B,BAXK2,MB, BSC, (KKX,KKT,KKZ, GINDPK 11 IGRLV* 1, V(K) IS GENERATED IN OEDEB CF K K VECTORS. CF IN OEDEB 1, V(K) IS GENERATED IGRLV* (6,360) NRITE SBITE(6,600) A,BO,ELECT,EBAG,IDCOB,KKPT,HAXK2,IDIR,BAXB2,HSTA, A,BO,ELECT,EBAG,IDCOB,KKPT,HAXK2,IDIR,BAXB2,HSTA, SBITE(6,600) IF (1GRLV. NZ.0. IKE. kSPlk.EC- 1. ANE. IH. EQ. 2) NRITE (6, 395) (6, 2) NRITE EQ. IH. ANE. 1. kSPlk.EC- IKE. NZ.0. (1GRLV. IF IF(IGBLV.NE.0 .AN D.NSPIN.EQ .2) .2) XIND-9 D.NSPIN.EQ .AN IF(IGBLV.NE.0 KEND=NKPI IF(KENE.EQ.O) EQ.1) NRITE(6,370) ND.IH. D.NSPIN.Eg.1.A .AN IF(IGBLV.EQ.0 GENERATE INDEPENDENT RECIPROCAL LATTICE VECTOHS. LATTICE RECIPROCAL INDEPENDENT GENERATE (IGRLVO.NE.O) IF IF (IGRLV.NE.0.AND.NSPIN.EQ.2.AND.IU.EQ.2) BRITE(6,405) (IGRLV.NE.0.AND.NSPIN.EQ.2.AND.IU.EQ.2) IF IF (IGRLV.NE.O. AND. AS FI N. EC- 1) KIND*3 1) KIND*3 EC- N. FI AS AND. (IGRLV.NE.O. IF GEKEBATES V (K) FBOR KST TO BEND IN STEP KIKCH. IN STEP BEND TO PBIHTED. KST BE OF TO V(K) (K) IS FBOR HO THE V KPTPKT GEKEBATES CALL FCEXC1(AX,AX,AZ,2£18, NHGSI2,£BG5IZ,HSBHOR, t ,RDIEC,0 ,NSPIN, ,NSPIN, ,RDIEC,0 t NHGSI2,£BG5IZ,HSBHOR, FCEXC1(AX,AX,AZ,2£18, CALL STOP NE.O) IF (IGAUS. IF (IGRLV. NE.O. AkC. AS PIN. EQ.2. ANC.IU.EQ. 1) 602) 1) NRIT£(6, ANC.IU.EQ. EQ.2. PIN. (6,403) AS EQ.2) AkC. BRITE NE.O. AND.IH. EQ.2. (IGRLV. IF NSPIN. AND. (IGRLV.NE.O. IF 1GRLV=0, V(K) IS GENERATED IN CBEEE CF INCREASING B**2 B**2 INCREASING CF IN CBEEE V(K) IS 1GRLV=0, GENERATED 1 KIRCE* + (NKPT) KST-KSQ 0-AND.IGRLV.EQ.O) (KST.EC. IF IF (IGRLV. NE.O. AND. NSPIN. EQ. 2. AND.I H. EQ. 1) BRITE (6, 600) (6, 1) BRITE EQ. H. AND.I 2. EQ. NSPIN. AND. NE.O. (IGRLV. IF IF(IGRLV.EQ.O.ANE.KSPIk.EC>1.ANE.IH.EQ.2) NRITE (6,375) NRITE IF(IGRLV.EQ.O.ANE.KSPIk.EC>1.ANE.IH.EQ.2) IF (IGBLV.EC.0. AND. kSPIN. Eg. 2. AND. Ii. EQ. 1) BRITE (6,380) 65) 1) (6,3 Ii. BRITE ITE EQ. 2) AND. Eg. 2. NR Eg. ANE.IH. kSPIN. EQ.2. ASPIk. AND. AND. (IGBLV.EC.0. IF EQ.0. IF (IGBLV. IF (IGRLV. NE.O. AND. NSPIN. EC. 1. AND.IH. EQ. 1) NRITE (6, 390) (6, 1) NRITE EQ. AND.IH. 1. EC. NSPIN. AND. NE.O. IF (IGRLV. IF (IGRLV. EQ.O) K1ND*2»ASFIK EQ.O) K1ND*2»ASFIK IF (IGRLV. 1,2X,'EXCH DOHN',4X,'D0*,9X,'D1',9X,*D2*,7X, 1,2X,'EXCH CIVXCR) 1,51,'H DCHN',6X,*D0',9X,*D1',9X,'D2',7X, 4'SOM VXCKPK',/) 4'SOM 2'DEN DP*,5X,'DEN DCHN',3 X,»KX',1X,• K1•,1 X,'KZ*,1X, • NO*,1X, • X,»KX',1X,• K1•,1 X,'KZ*,1X, DCHN',3 DP*,5X,'DEN 2'DEN CIGELVO,ISUKT,lfiCILD,NSFNT,BFTIK,NRPHl,NDlRC,NFILCN CIGELVO,ISUKT,lfiCILD,NSFNT,BFTIK,NRPHl,NDlRC,NFILCN C,•SUB VCKPK') C,•SUB 3'CHECK DENS' ,2X,'SUH VEK',2X,'SUB V EKPK',2X,*SDB VCKPK',2X, EKPK',2X,*SDB ,2X,'SUH V VEK',2X,'SUB DENS' 3'CHECK 2*DNDP(K) ' ,4X,* DEKSITI (K) ' , IX,'KX', IX, *KX*, 1X,*KZ', 1X,'NO',3X, 1X,*KZ', *KX*, IX,'KX', , ' IX, (K) DEKSITI ,4X,* ' HXCKPK',/) 2*DNDP(K) SUB VCK•»4X,' DENS',2X,'SUB 3'CHECK C'SOB VCKPK', 2X, ' SUN VXCKPK',/) 2X, ' SUN VCKPK', C'SOB CCA CCA C,6X,' H DOHN',4X,'DENSITI',3l,'CHECK C,6X,' DENS',/) H DOHN',4X,'DENSITI',3l,'CHECK , IX,'NC, IX,'KZ* , IX, ',K*• JX,'KX',1X,» (K) 2X,'CENSITX , (-2/3) * 2 'DEN** 3'CHECK DENS',/) 3'CHECK C•DENSITE(K)' ,2X,*CBECK DENS•,3 X,' SUB VCK',3X,'S0B X,' BXCKPK',/) DENS•,3 SUB ,2X,*CBECK C•DENSITE(K)' DENS',/) 3X,'CHECK DENSITY•, 2X,* DCHN•, X,'EXCH G,3 C'DENSITX (K) *,2X, 'CHECK DENS' , 1 X, 'SON VEK' ,2X, • SU (! VEKPK',2X, (! SU • ,2X, VEK' 'SON X, 1 , DENS' 'CHECK *,2X, (K) C'DENSITX 402 FORMAT(72X,'i)J*,SX,'D4*,7X,'SDM VEK*,3X,»SUH VEKFK',3X VEKFK',3X VEK*,3X,»SUH 402 FORMAT(72X,'i)J*,SX,'D4*,7X,'SDM *) (72X,'D3*,9X,'D4',9X,'SDM HKPK OP' 403 FORMAT 405 FORBAT(3X,'K2',1X,'COOLOUBB*,2X,'ENALD',5X,'B',9X,*N 395 FOBB AT (2X,'K2' ,11, 'COU LOUBB (K) * , 2X, • ENALD (K) * , NX ,' N (K) ' ,7X, ' ,7X, (K) ,' N NX , * (K) ENALD • 2X, , * (K) LOUBB (2X,'K2',11, 'COU AT 395 FOBB 400 FORMAT (3X,'R2',1X,'CCD ICO ME', 2X,•EH ALE*, 2X,• EXCHANGE', 4X,' EXCH OP' EXCH 4X,' 2X,• EXCHANGE', ALE*, 2X,•EH ME', (3X,'R2',1X,'CCDICO 400 FORMAT 360 FOB BAT (1 HI) (1 BAT 360 FOB , •,4X B(K) , 'ALE(K)' NX,' , 2X,•IN K2*,1X,•CCULOOHE(K) X,* (2 375 rORHAT OP* 385 FORMAT{2X,'K2*,1X,'C0D10UH£',2X,*EBAID',6X,'N',8X,'H 390 FOBHAT(2X,*K2*, IX, 'COULOOBB(K)•,2X,•EHALD(K)•,1X,• EXCHANGE(K)',11, EXCHANGE(K)',11, IX, 'COULOOBB(K)•,2X,•EHALD(K)•,1X,• 390 FOBHAT(2X,*K2*, 380 FOAM AT(2X,*K2', IX,'COULOOBB•,2X,'ENAID',3X,'EXCHANGE',3X,'EXCH DP' DP' AT(2X,*K2', IX,'COULOOBB•,2X,'ENAID',3X,'EXCHANGE',3X,'EXCH 380 FOAM 370 FOEBA1 (2X,' K2* , 1X, • COD IOUBB (K) ', 2X, • IN ALD(K) ■ , IX,' EXCHANGE (K) ',11, (K) , EXCHANGE IX,' ■ ALD(K) IN • ',2X, (K) IOUBB COD • 1X, , K2* (2X,' 370 FOEBA1 wuu uuuuuu 95 HO* , 1X, , HO* X. • X. 1, NFLDIH.KI ND, CB1) ND, NFLDIH.KI 1, b , b £SN=PHB1GA. E10LA. FCOF E10LA. £SN=PHB1GA. ,/) * DEBS DEBS 2— 1 DO 570 NPT=KST,ICEND#KINCH 570 NPT=KST,ICEND#KINCH DO (NPT) KZ~KKZ 450 TO GO NE.O) (K2. IF K¥*KKY(NPT) 0 42 TO GO ENDCN*(2.£0*ELECT-ALPHA)/EETA1+1LPHA/EETA2) NLDBCEND*((-2 GE MB E*M B(HPT) E*M MB IF (NT (I) .L£.0. DO) .L£.0. G0T0425(I) DO) (NT IF KX=KKX (HPT) KX=KKX (NPT) K2-KSC DO *BFT1N 425 1*1 DO CO*6.DQ*HT(I) 1+RHORV1(II)«CO 1*Cfi CB GC0DL=CB1*CD2NS*(-4.D0*PI/3.D0) 430 I=1,BFTIN DO CALCOLATE THE FOUBXEB COIHICIENT V (K=0) V COIHICIENT FOUBXEB THE CALCOLATE BIC=0.DO GU HDN* EIEDN/OHEGA GUflUF*£LEUP/O0EGA HDN* GU Cl II(CBO,1#NBPNT,BK,DRGAUS,BH01#B,0#NBPNT#KIBD,CR1) Cl FILOI KS RK*DSQET RK*DSQET (BS) DO 429 DO K*1,KINC SDN*ELECT/ONEGA 460 TO GO IF (IGRLV. EC-0) 410 TO EC-0) CC IF (IGRLV. ,RFILO FLBHOB DBFILN, BK, (CBO.I.HFILON, FILC1 CAIL II*1 I I “II*EFTIN (I).LE.O. TC430 £0) GO IF (NT CO*6.DO*NT(1) FILC1 CA1I (CBC,1,MBGSIZ,BK,CBilGSZ,USRB0R,FNG51Z#1,NUGSIZ#KIND, IF (IGRIV. EC.O) GC TO 440 TO EC.O) GC (IGRIV. IF *CO HHOBHF(II) ♦ *1 II (K) *CRO CBC(K) 11*11+BFTIN TO 550 GO BK2* DFLOAT (K2) DFLOAT BK2* BS»BK2*AKB2 CA1CU1ATE THE FODRIEB COEFFICIENTS (K.GT.O) COEFFICIENTS FODRIEB THE CA1CU1ATE CAICULATE THE FODBIEB COEFFICIENTS OF THE EXCHANGE POTENTIAL EXCHANGE THE OF COEFFICIENTS FODBIEB THE CAICULATE 460 TO GO (IGBLV.NE.O) IF 2 *DEN UP1, 5X, "DEN DCBN * ,3*, • KJ* , 11, • K1* , 1X, • KZ" # 1 # KZ" • 1X, , K1* • 11, , KJ* ,3*, • * DCBN "DEN 5X, UP1, *DEN 2 3 "CHECK 3 "CHECK SCSI) 410 E2-NPT-1 NOB 410 E2-NPT-1 0 CONTI 42 425 425 CONTINUE 429 429 CONTINUE 430 CONTINUE 430 CONTINUE 0 CONTINUE 44 450 450 CONTINUE U U U U uuuuu nnn CONTI 452NOE S0H=2.D0*SDfl-JP451 nonnnn r>nnn 6 FORMAT464 CONTINUE462 (65X,5F11. 6) 6 CONTINUE460 EHBITE K2,GCOUL,GEHLD,GEXCH,GEXDP,GEZDN,SUH,BHOK ) 0 8 5 , 6 ( H I E VEKSOB, YEKPSB, VCKSOH,GHBITE VCKPSH IF (NSPIN.EQ. IF GC TO 1) 470 IF IF (NPT.LE.KPTPBT.Ofi.(K2/2CC)*200.EC.K2) G U H D N aS U f l D IF (NSPIN. N EQ. GC1) TO 451 CALL FILON(CRO,1,NRPBT,BK,EBGAOS,H0CB,B,O,NBPNI,KIND) IF(N PT.EC*619.CR.NPT.EC*620.OB.(K2/5C0)*500.EQ.K2) GEXDN-CBOGEXUPa (4)•CEXFE CRC(J) *CEXF£ GCOUELaGCOOLa SOtt*0.DO*PI/BS (SOff*B.DO*Pl-COl)/BS CALCOLATE THE FOUEIER COEFFICIENTS OF THE COOLCHB POTENTIAL. INTEGBATECHABGE EEOR B=0 TO THE ASSURED DENSITI**1/3 BADIDS IS OF THE TO EE HIGHEH-SE1IZ SPHERICALLY SPHEBE. SYBBETRIC- IF (IN. GO EQ.2) TC 462 CALL CFHYGF(1 .D-10) .DO,1 .5D0,I,CFHIG,1 E N = C F HCALL I S CFHYGF(1.DO,1.5D0, * A L P H X,CFHYG,1.D-10) A / B E T A 2 GO B=S0 B C A LCO L U L O H ( C H A B D N , S U HC D N A , L ECC S L , F A D C L T O ) B ( C H A B O P , S U B O P , R S , F A C T ) CALL FILON (CBO,1,NNGSIZ,BK,DBBGEZ,NSBHCB,BNGSIZ,1.NiGSIZ,KIND) CALL FILON (CHO, 1 ,NFILON,HK,DBFILN, FLBHOB,BFILD N, 1, NFLDIJI,KIKD) ANNB«DFLOAT(NNNB) GO TO 452 S U H aS U B D P * S D M D N GUMOP*SUBUP NBITE( E2,GCODL,GENLD,GEXCH,GEAOP,GEXDN 1) V E K P S B aVV E E K K P S S 0 H M 4HBITE( G = K2,GCCDL,GEHLE,GEXCfc,GEXOP.GEXEN.GOBUE,GUBDN,G0H C V ) 0 1 0 E U K E S L O * B S * U G B C C O I L G E X C H = CB » 0 H { 2 O )* K C sC E B X O P A < 1 ) * C D E N S G E 6 L D aC EEH*Eli*CFbIG»(- B D * E H 2. CO * ELECT-ALPHA)/EET A1 1»-BS/ (4. DO* 3E1A2) G0HaSUH F0UB1EH COEFFICIENTS ABE GENERATED AS FUNCTION K**2- CF ? C K P S H = V C K P S R * G C O D L * S 0 H V C K S D B aV C K S O B * G C C U I NHEN GEXCH(KS)=VIC(KS> IN.EQ.1, NNbB*NKB X«-RS/ (4. DO* 3ETA1) C O U L OPO H T E N T I A L GET AND BO DP BO (K) DN (K) THEN V(K) FCE THE DSN=PHBAGA.EIOLA.FCOF 97 DSNxPH3AGA.DICLA.FCCF 1 *KY* *KY* 2 HBITE (11) K2 ,GCCUI, GEN LD, GEE CL, GEXOP, GEXEN, GDBUP, GUBDN, GUB GUBDN, GDBUP, GEXEN, GEXOP, CL, GEE LD, GEN ,GCCUI, K2 (11) HBITE WHEN IN.EQ.2, GEXCH(KS)*H(KS) WHEN POTENTIAL ONLY POTENTIAL (10)K2,GCCDL,GEHIE,GEXCH,GUH HBITE HBITE (1) (1) EL K2,GCCU1,GEHLE,GEXCH HBITE GCCU VEK5U.1* VEKSUB* HXCPSB=HXCPSB+GEXCH*SUB VEKPSB*VEKP5B+GC0UEL*SUH VCKSUHxVCKSOHtGCCDL FILE 2 IS USED FOB CALCULATION OF THE TCIAL ENERGY USING TEH TEH USING ENERGY TCIAL THE OF CALCULATION FILE 2 FOB IS USED (2) HPDNSB BDN K2,GC0UL,GEHLD,GEXCH,GEXDP,GEXDN + HBITE SB HPUP HPU?sa=NPU?sa*GEXop*sonup HPi/NSn= 0SNsPHBAGA.D20Ll.rC0P 22= (BK2/BK) **2 11=12*22 14=12*12 24=22*22 C41= X4 +Y4+Z4-Q.6D0 C6 1=X2*Y 2*Z2+C 41/22.DO-1.DO/105.DO C8J=14*24+Y4*I4 + Z4 *2 4-5.6D0*C61-21G.E0*C41/143.DO-1.DO/3. DO CUBIC (1) = 1. DO CUBIC (2) *C4 1*BN041 •-* SIGN COMBS FBOM 1**6. CUBIC (3) =-C6 1*EMC61 CUBIC (4) =C81*BN081 DO 520 ND=1,NDIKC II=ND DO 520 K=1,KINC CfiO(K)=CBO(K) +ABO(II)‘CUBIC(ND) 11=11+ NDIBC 520 C0NI1NU2 CALCULATE THE FOURIER COEFFICIENTS OF TEE CHABGE DENSITIES FOB BOTH SPINS . IP (KS2.EC.K2) GC TC 540 K52= K2 X=-fi S/ (4.D0*3ETA2) CALL CFHTGF (1.DO,1 .5D0,1 ,CFHYG,1.D-10) EN=CFUYG*1LPUA/BETA2 X»-BS/(4.D0*3ETA1) CALL CPU7GP(1. D0,1.5D0.X,CFEYG,1.D-10) EH=EB +CPHIG*(-2.D0*ELECT-ALPHA)/BETA 1 G£KLD=CEUD*Eif CALL CCULOB(CdABUP,SDHUP,RS,FACT) GUMUpsSUBUP IP (NSPIN. EQ. 1) GO TO 530 CALL CCOLOH (CBIRDN.SDHLN, RS, FACT) GDMDN=SDMDN GCOUL=((SUMUP+SUBDN)*8.D0*P1-CC1)/BS GCOUEL= (SUMUP+SUBDN)*8.D0*FJ/BS GO TO 540 530 GUMDN=GOMUP GCCUL= (2. D0*5U (IUF*8. DO *PI-CC1)/BS GCOUEL=2.DO*SOHUP*8.DO*PI/BS 540 CONTINUE 550 CONTINUE IF (NSPIN.EQ. t) GO TO 560 BU0K=CB0(1) +CDENS GEXCH=CB0(2) +CEXFA UEXUP=CBO (3)*CEIFE GEXDN= CBO(4) *CEXFE GDSPA= CBC (5)+CEXFE GDSUP=CBO(6) *C£XFE GDSDN=CBO(7) *CEXFE GDSUD=CRC(8)*CEXFE GD5DU*CitO (9) *CEXFE MNBB=NNB ANNB=DFLCAT (.IN NB) 100 DSN-PH8AGA.DI0LA.FC0F XF (NPT.LE. KPTPBT.OB. (K2/100)*100.EC-K2) 1NRITE (6,580)K2,GCOUL,GEHLD,GEXCH,G£XUP,GEXDN,GDSPA,GDSDP,GDSDN, 2GUSUP,GDHDN,KX,KI,KZ,NNS ,BHOK XF (NPT.LE.40) 6VBXTE(6,584) GDSUD,GDSDU XF (IH- BO-2) GO TC 552 HBITE (1) K2,GC0UL,GEHLD,GEICH,GEXUP,GEXDN,GDSPA,GD5UP,GDSDN,GDSUD, &GDSDU,GUHUP,GO EDN,KX,KX, K2,NNB IF (K2.EQ.0) GO TC 552 VEKSOH*VEKSUB*GCCOEt*ANNB VEKPSB=VEKPSH*GCCU EL*(SU HU P+SDHDB)*ANNB VCKPSH=VCKPSH*GCCUL* (SUHOP+SUMDN)*ANNB XF(K2.EQ.O.OB. KPT. EQ-2-OB.KPT.EQ.619.OB.(K2/50 0)*500.EQ.K2) 6HR1TB (6,586) ?EKSUH,?ERPSH,TCKPSB 552 CONTINUE XF (IH.EQ. 1) GO TC 570 HR1TE (2) K2,GCCUL,GEKLE,GEICR,GEXOP,GEXCN,GDSP1,CDSUP,GDSDN,GDSDD, 6 GD50U,GUflUP , GDHDN, KX,KT,KZ,N NB XF (K2. HE. 0) 6NKPKSH*HKEKSB* (GEXDP*S0H0P*GIXDN*S0HEH) * ANNB XF (K2.EQ.O.OB. NPT.EQ.2.OR.NPT.EQ.615.0B.(K2/500) *500-EQ-K2) t NRITE (6,586) > KPK5B GO TO 570 564 FOBBAT(71X,EI0.4,2X,E10.4) 586 FOB9AT(90X,3F11.6) 560 GUB^GUBUP+GUBDb BHOK*CBO(1) *COENS GEXCH=CBO(2) *CEXFE GDSPA*CB0(3)*CIIFE NNNB*NNB ANNB=DFICAT (NNKB) IF (IH.EQ.2) GO TO 565 HBITE(1) K2.GCCU1,GENIC,GEXCH,GCSPA,GDB,KX,KY,KZ,NNB XF (K2.EQ.0) GO TO 563 »EKSUH*?EKSUH*GCOUEL*ANNB 1EKPSB=VEKPSfl*GCCUII*2.DO*SUBUP*ANNE BCKP5n>TCKPS9*GCC0L*2.DO*SUBUP*ANNE FXCPSH*trXCPSB*GEXCH*2.D0*SGRUP*ANNE 563 CONTINUE IF (NPT.LE.KPTPBT.OB.(K2/100)* 100.EQ.K2) 1HBITE(6,590)K2,GCOUL,GEhLD,GEXCH,GOSFA,GDE,KX,KI,KZ,NN3,RHCK, 2 VEK5UB,VEKP5H,VCKPSfl,VXCC5B 565 CONTINUE IF(IH. EQ. 1) GO TC 570 IF (K2. EQ.O) GO TC 566 VCKSUH*VCKSUH*GC0UL*ANN3 hXCPSB-HXCPSB*GEXCU*2.CO*SUBUP*ANNE 566 CONTINUE HBITE (2) K2,GCOUL,GEBLD,GEXCH,GDSPA,G0H,K1,KX,KZ,NKB XF (NPT.LE.KPTPET.OB.(K2/100) *100.IQ.K2) INBITE (6,590)K2,GCOUL,GENLD,GEXCH,GDSP1,GOB,KX,KX,KZ,NNB,RHOK, 2VCKSUfl,NXCPS.T 570 CONTINUE 580 FORBAT (1 X, 15,2 F8. 4,6 (1 X,E 10. 4) ,413, 11,210.4) 590 FORBAT(II,14,5111.7,2X,413,1X,6F11.7) HBITE (6,600) A, RC, ELECT , EH AG, 1CCUE, NKPT, HAXK2,IDI(1, HAXB 2, NSTA,IGRLV £,ISOBT,IRCELD,NSPNT,NFTIN,NBFNT,NDIRC,NFILON,NSGSI2 600 FOBBAT(IX,//,IX,'LATTICE CONST**,F10.5.3X,*B=*,F10.S ,3X,«ELECT NO* 101 DSM-PHEAGA. C10H. PCOF 1*,F7.2,3X, • HAG BO*' ,F7,4, 3 J , -IDCUB- • ,1 5 .//, 1X, 2'K NO-',18,3X,'K2 BAX-*,15,3X,*B BO®«,18,3X,*B2 BAX-*,IS,3X,'OBBIT 3 ALS NO-'alSflXf'GEK RLV-•,12,3X,'SORT*',1 2 ,//,1 X,'H D1V=',15, 43X,* B PTS=',I5,3X, 'HONSPHERICAL PTS- » ,15,JX , * GAOS PTS=»,l5,5X, 6 'DIRECTION®• ,15,3X , *NF1LON-*,1 6 ,3X ,* NHGSIZ-',1 5 ,/ ,1H 1) IP (IB.EQ.2) GO TC 603 INC FILE 1 IF (IVXCH.EQ.2) IB = IH*IDO IF(lii. EQ.2) GO TC 307 GO TO 300 603 CONTINUE IN D PILE 2 GO TO 300 610 STOP EN C SUBBOU1IHE VXCHS1(UP,DN,DENS,EXPA,EXUP,EXDN,DVUDBD,DVNDEN,DVODBN, 1DVNDBU,DVPADR,CVUDBH,DVNDRH, DVUDH, DVND!1,DEL0,DVI,DVTA) C REVISION OP SUBROUTINE EXBATH, DECEMBER 1979 IMPLICIT BEAL*B(A-H,C-Z) COHHON/TOTLE/IW P{Z) = (1. DO-Z—3) -DLOG(1.D0+1.D0/Z) ♦ Z/2.D0-Z-Z- 1.DO/3.DO DEHS-DF+DN PI-3.141592653589793D0 ONETHC-1.DO/3.DO AA=1. DO/2.DO --ONETHD COEF=4.D0*AA/(3.D0- (1.D0-AA)) ONEFOfi-1.DO/4.DO FOR1HD-4.DO/3.DO CP-0.04612D0 BP-39.7DC CF=.0262BD0 BP-70.600 BS = (3.DO/(4.DO*P1»DENS)) **ONITHD UPX--2.D0* ({9.D0-P1/4. £0)--ONETHD)/ (Fl-BS) ECP--CP-F (ES/RP) ECP—CF*F(BS/HF) UNC-COIP* (ECP-ECP) OCP—CP-DLOG (1 .DC + BP/H S) UCF=-Cf *DLOG (1. DO+BP/BS) A-UPX-UNC XC-UCP-UCP-FOBTHD*(ECF-ECP) B= (UCP-AA-OCP)/ (1. DO-AA) C=TC/( 1.D0-AA) XUP-OP/DENS XDK-DN/DEKS IF (IB. EQ.2) GO TO 10 X4-XOP--FOETHD -XDN--P OFTliD CQ-B-C-X4 QA*(2.B0*X0P|*-ONETHD QB-(2.E0*XDN) --ONETHD QC-AA-QA QD-AA-CB EXUP-Cfi*A-QA EXDN-CQ-A-QB EXPA-A-B-AA-C PLX- (X4-AA)/(1.DO-AA) DVT-28.DO-ON2THD*(ICP-ECP-6.DO-(DCF-UCP)/7.D0)• (FLX*AA/(1.DO-AA)) DV1 - DVT-CP-B1-FLX/(FF-FS)-CP-BP- (1.DO-FLX)/ (FP-HS) 102 DSN-P8BAGA. LlOLl.f COP DVT=ONETHD*DVT/DENS DVUDRU=DVX*OjJETHD* A* CA/U F*B- E0*O NEIH E*C*QC/DENS LVNDR»=DVT*OKETBD*A*QB/DN*8.DO*ONETHD*C*CD/DENS DVUD6N=DVT«-FJRIHD*C* (QC+CD) /DENS DVNDBU=DVUDRN DVPADB=DVT*B.DO*ONETHD*C*AA/DENS*ONETBD*A/DENS DVUDHH=0.5D0*IDVUDRU*DVUDRN) DVNDBH=0.5D0*(D¥NDFD*DVNERN) DV0DH-0.5D0* (DVUEB U-DV ODfiH) DVNDII=-0.5D0*(DVNDBN-DVUDRN) DELO=O.SDO*(DVUEBB*D¥NCRU) DEL1=0.SDO*(DVODBH-DVNDBH) DEL2*0. 5D0*(□¥ UDH-DVNDfl) BETUBN 10 CONTINUE CB«AA*(DCF-UCP-7.DC*ONETHD*(ECP-ECP))/(1.DO-AA) PB =(ONEFCB*DPX*0AC+GB)/AA GB=OCP-ECP-U»C-GB BUP=Pfi*XUP**ONEThD*0B NDN=EB*XDN**ONETHD + QB MPA“PB*AA*QB DGDRS=AA*(CF*BF/(RS-*BF)-CF*RF/(BS*RP)-7.E0* |ECF-ECP-UCF*OCP))/ e (1.D0-AA)/BS DP DBS—UPX/ (a. D0*AA*BS) *4. DO* (ECF-ECP- DCF* UCP) / (I. DO-AA) /HS* GDCDBS/AA DQDBS=CP*BP/B5/(FS*BP)-3 .DO* (ECP-OCP)/BS-4.DO*AA*(ECF-ECP-DCP+UCP) £/( 1. DO-AA)/BS-DGESS DVD1 = - (DPDBS*XUE**CNETBD*DCDFS)*BS*ONITHE/DENS DVD2*- (DPBBS*XDN**ONETHD*D0DtS) *RS*ONETflD/DENS DH DDB0=DVD1*PB *XDN *XUP**(—2.BO*ONETHD)*0NE1HD/DENS DBNDRN*DVD2+PB*X0E*XEN** (-2.DO*ONEXHE) *ONIXHD/DENS DH UDBN*DVD1—PB *XDP**0NETHD*0 HETED/DENS DH NDBU=DVD2-PR*XEN**ONETHD*ONEIHE/DENS EH PA DR— (DPDBS*AA*DCEES) • BS*ON ETBE/DINS EXPA*HPA EXUPeHDP EXDN*NDN DVDDBU-DH0DB3 DVNDBN*DBNDBN DVUDRK=DHUDRN DVNDB0*JHNDE0 DVPADB=DN PAD3 CVT*DVD1 DVXA-DTD2 BETUBN END SUBROUTINE POEICl (AX,A Y,AZ,1DI N, NBPNT,B,HHOB,NEIBC,NDTOL,IGNAT, CNSPIN,IVXCB) C C CALCULATE THE KDBIC HABHONICS EXPANSION COEFFICIENTS OF THE C FUNCTIONS OF TBE CBABGE DENSITIES. C GAUSSIAN TIPS ORPITALS AFE USED. C £••••**#**••*•**•••**••**•*•••***•*• IHPLICIT BEAL*8 (A-F,H,O-Z) D1 HEN SION AX (I DIN) ,AX (IDIN) ,AZ(IDIB) ,BUCR (1) ,R (1) D1BENSIUN CC(4,«).D(4) COHHON/VK0/C (7,14),EX(7,14),FACTO(10),1B(7),NOBB(7) ,NSIA non n n r> n nnn n o n 4 0 FOfiAAl (1X,'A*(, ,3F 10.5,')• ,5X,'K01*', F10.5,5X,•K41=•,F10.5,3X,•K61 F10.5,5X,•K41=•,F10.5,3X,•K61 ,5X,'K01*', 4 FOfiAAl 0 10.5,')• ,3F (1X,'A*(, 0 OBT /,4, ',X'DENSITY',10X,'KDB1C FOBBAT(60 ,9X,' ,//,14X,* F' HABA. X I COEF. POB THE P HBITE20 (6, 30) BX ( X),BZ(I) ,(CAAT(I.J),SY(I) ,JS1,NDTC1) 0 FOBBAT(1X,aA*(*,3F10.5,')',4115.5) 30 0 8H0B70 DO (I)*0. 0 CONTINUE50 CONTINUE10 ',F10.5,3X,'K61*',F15.5) * ( C A B A B A G B E TEX 1 C C H A N GPC E T E N T 1 A L ' ,/) IF (NSPIN.EQ. GC TO1) 170 COBAUli/XCONST/A,PI,SIXPl,OmHD,lMTHlV,IPUK.IGAOS DENSUF*0.D0 I K 6 T * N F P N T * N D I B C 1.MDTOL 201s 00 CALL GAXS(HOTOI,CBAT,CC,D,DET,4) CALCULATE THE XNVEF5E CF THE BATBIX. Cfl ATCBAT (I,3)-C61*CN061 (I,2)*C41*CNC41 C81*X4*X4 C6+ X2*Y2*Z2+C41/22.DO-1.DO/105. Is Y4*Y4 +00 Z4*Z4-5. 6D0*C6 1-210. £0*C to o n *a n n O non 3 CONTINUE230 1 CONTINUE210 3 CALL233 VXCB51(DENSUP, DENSOP,DENS,CX,UP,DN,D1,D2,D3,D4,D0,CA,C8,CE, 200 IF (IBB.NE.O) IF 200 GC TO 220 2 sun*sun*psi*psi 220 3 CONTINUE237 5 7E T1,47.,(XE04,( ,E9-3)) 70EH150 AT(1*,14,77.3,9(1X.E10.4),2(U CONTINUE130 7 CONTINUE170 4 CONTINUE140 190 CONTINUE 180 BNOfi(I)=0.DO 6 CONTINUE160 CCF,DEI0,DEL1,DE12) 13=12*1KBT B=B(I)-1 - ) I ( 1B&=1B I2=IK6♦IKHT £AXs RA2*EX (I,J) D E N S U P = D E N S O P * S U E * C H A R U P ( I ) • C O N S T GO TO 220 1KR= *NHPNT*KB ) 1 - 1 ( 1,NDIEC s DO I 240 S H O E ( 1 2 )= B U O R ( 1 2 ) * C N A 7( 1 , N D )* U P PSI=0.LJ PS1=PSI*C(1, J) DO 2 10J= 1, N IKKTOL=lKBT*3 BEXs J)*HA**IEB*D£XP(-EAX) (I, C S1,IDINDO J J 220 IIsNOBB I) ( 0.DO S0Bs U P = D E N 5 U P * * 0 N E T H D » S I X P I NSTA , 1 s 1 0 3 2 DO Z = B ( K E )* H Z ( N D ) DO ND=1»NDTCI 250 KR=1,NBINTDO 270 DO I=1,IKHTCL 180 B H O R ( I K B ) = E H O B ( 1 K B ) * C I 1 A T ( 1 , N D )* D E N S SsPS1*BEXPSIs DO GOIF TO (BA.LE.0.0) 20C HA-DSCET (BA2) D E N S U P = O . D O IF (EAZ.G7.25.D0)GO 190 TO BA 2s (AX**2* {JJ)-I) (AX(JJ)-X)•*2♦ (A Z **2(JJ)-Z) C 0 S D E N S * * T M T H I V * 5 I X P A / 3 . D Q F A E A H A G 2 N T 1 C . RETUEH SHCB(19)=HHOR(1 )*CHAI(I,NE) 9 *D4 X = B ( K F ) * B X ( N O ) NRITE D E N S = H U O B ( K B ) * C N 0 0 1 S=B(KR)*BY (NO) S=B(KR)*BY DENSSDE NSUP*2.DO 190 Js Js 1, 190 N GC TO 237 IF(IVXCB.EO.2) GO TO 233 610 KP,B(KR>, DENS,(6,150) CX,UP.DN,DO,D1,C2,D3.D4,DVT.DVTA DSN=PBBAGA.DI0LA.7C0F 105 n ft ft n ft n o n K> KJ Kj KJ m t O o ci ft o © 0) I 1 9 1 0 X M W 0 © X 0 H 0 H ft ft ft ft ©O H H a tfi ft * in ft W f t f t 3 tftO0OO'AN03O999tflO B C l_i o KJI N N * * 0 0 0 C 0 0 0 OS!o o o 3 C 3 in e z S8 Bh O O k m Z M IK ^ K * H KjH Z MM M N X X 3 m w m z 3 3 9 3 X 0 Z 0 3 0 © © X Z H z z z o * ^ M n X K HKjBO HH If i u>o S V^ K 0 0 nintA ^ a 0 m 0 z In 0 3 9 3 58 w ft Vl W ft M 3 a x X W X X 3 0 O B 9t0O9tnM©olr O M 0 ft O M O * 0 X0 O C O 0 0 6) o in x 9 o o o Z Z tf) M i/i to e Z O m mw II H Mm (ft 0 Z (Am k 0 •0 9 * 10 ■ 0 K m Z * 0 0 • X > V 0 NI* Z Z 96 in in n ft H 0 ft * C W Z Z M N Cl I * KJ Lfld I WKO H M C.m©*** H ♦ 0 0 0 H |T * r c k» »W\ H H » M 3 0 3 ft X s a c <-> * iikcn*« a «. ii to X m o Cj H I K X tO 50 » C © 0 in 6) H X K «3 O C • X * V) X O fJ ft 0 C X 0lu* . H w ' -* • t \ • Z • a K ft ft Z z z Kj M 9 3 H w w * M i«* e v e «* • * ♦ OOH X* K ft M © 0 0 3 0 9 X X X ft M W H • M a . 1Z ■ > —' • i m ♦n n o © © «* * O' W X C\ H Z K \ 9 9 N to x M ft H Z X to in M © KJ I H In N o o o * ■ N • • m f t f t f t K EC 3 3 0 0 3 Z 3 3 o * * « — K © X »0 tt•xi z tj* O mi £:l o H O ^ m O X ft 9 Z 3 • X n © c, I* 0 0 0 •w w a • \ \ X H B M * 9 H © S. I n o w o « Z 0 © Cf 9 ft* ft A Z 9 o a a < 0 o * * W * K 6) O O * 3 «« W—. 9 0 9 3 Z 0 o Kj I M* 0 O ♦ • 0 9 ^ 0 3 0 n «* 0 o X -* H 0 • M »*"" Z I H Z © B m • n X O X X MO * ft* X 0 X W 9 « 9 Z ■ in © 1 K 0 z ft m to 3 e z » M to o 0 i w ©* m hn Mm x Cm * EC * f t X w B« H o z • en H X w 3 H O 9 0 I Ci • k« 3 0 W 3 X 3 X © •a* 1 Cl * « w © M* * o* 3 ? 9 ft H to 1 \ • O * ft «»M 9 M N 9 n I n 0 • N. -» O erc 3 EC Z"- X 9 (A I w X w © — Z 9 w ft 3 X I z M 0 0* • 0 0 a n 9 © I li • X a 0 3 < 3 3 X © 0 I 0 3 0 3 N ^ w M I X Kj ©M ft # P ft 0 9 I © ♦ * XX 3 O I 9 O * 3 M I © 3 9 Z SO I 9 © X ft -* 3 M 61 H K I « K X 3 ©—’ 9 tn I ft t* H U X 0 * 3 ©M 3 IS * ft 3 0 0 H X I W Z 9 Z 9 l£ M 0MM « (A M a x x H * as • X H N K K 3 « KJ * M * ft 61 o O © 3 0 3 0 a 9 » Z IA N in 3 M « 0 M © £ X X 3a o K 3 © Z0 (A O C7» non 2 FOBSAT120 DENSITY*»t6X,' ,10X,•r•,10X,•Z, ^//■,6X,•R, ,10X,*X, #//) 1 CONTINUE110 0 CONTINUE100 7 COKTINOE97 FORHAT80 (3F9.7,1X,4F13.7) CONTINUE60 CO40 i HUEm 0 FOEMAT(1X,4F10.90 5,4120.6,15) FOW1AT70 (41X,3F20.S,15) 0 SUft«SOfl*PSI*PSI50 93 W IE 78 ) X,Y,Z,DENS,DENSOP,DENSDN.NT(KB)CWHITE(7,80 ) £CF,DE10,DE11,D£12) B2,X,Y,Z,DENS,DENSUP,DENSEN,NT(KB)GNR1TE(6,90 ) IT (IPON. NE.O) DO DO GO TO 97 C X sD E N S * * O N £ T E E * S I X P A IF (I G A US. EQ.O) GO TO 60 N*ROBE (I) 1BB*IB(I)-1 IF (IGAOS.HE.0) BBITE (6,120) P i B B B A G E N T I C . B H O B ( K R * 6 * N R P N T ) » C H F T IB N * B D 2 O f i ( K B * 3 * N B ? NB T U ) " C O H F E X I N ( K ' D E N + N B P N T ) - C 8 F T I N * C X CALL D E N S = D E N S D P * D E h S E K D E N S D N - D E N S D N * E B E N D E N S U P - O . D O HETOBN S H O E( K B * 4 * N B P N I ) * C ( I F T I N * D O D1«DENSUP**TWTH1V*5IXPI/3.D0D O = D E N S * * T N T i J I V * S I X P A / 3 - D OIF (IVXCB.NE.O) 93 GO 10 IP (1G1US.NE.0) D H D NDBUP=SUH»CHiROF(l) = S U I 1 * C H A *C0NST R D N ( I ) *C O N S T 50(1-0. DO DO 1701-1,NSTA B 2 - X * X + Y *Z-IfiZ(KR) Y * Z * Z *COfl I*JEY(KB)*C03 BHOF (KB*5*NRPNT)-C6FTIN*E1 DHOB(KB*2*SBPN1)-C1!FTIN*DPB U C E (B K H P ) » 0 C R H V F T 1 I(K N * R D ) = E C N H S F T I K * D E N S * B 2 D N = D E N S DUP»DENSUP**0NEIHD*SIXP1 N * * O N E l U D * 5 I X P I X«1RX(RR)*C0R BHOB(KF*8*NBPNI)=CBFTIN*E46 H C B ( K R * 7 * N R P N 1 ) ■ C H F 1 I N * D 3 DBF,DEOP,DSDN.l ) NBITE(6,70 D B P S D K U P + D B DD N E N P S U P = i ) t N S 0 P + D B U P DO KE*1,NHPNT 190 D2-DEN SDN **TNTHIV*SIXP1 DO . 3 / 6 JJ=1,ID1H 160 VXCB51 (DENSOP,DENSDN,DENS,CX,OP ,DN,D1,D2,D3,D4,D0,CA,CB.CE, DSN-PHBIGI.DI01A.FC0F 107 108 DSN-PBBAGA. DIOLA.FCOF DSN-PBBAGA. psi * psi * som - SDBBOOTINE FILC1 (ABC, NCIEC,NEPBT,BK,EBF1LN,BH0P,BF1LCN, NZEBC,NDIB, NZEBC,NDIB, NCIEC,NEPBT,BK,EBF1LN,BH0P,BF1LCN, (ABC, FILC1 ENE SDBBOOTINE PS 1=0. DO 1=0. PS DBUF,1 NBITE(6,180) PSI- PSI* C (1, J) (1, C PSI* PSI- TO 187 GO 8*2- 8*2- (*I(JJ)-X)**2+ (AT(JJ)-Y)**2«(AZ(JJ)-Z)**2 BA=DSQBT(BA2) EAX-BA2*AX(I,J) UP-DEASU P**ONETHD*SIXPI P**ONETHD*SIXPI UP-DEASU DO—D£NS**TNTHIV*SIXPA/3.DO BU 08 (KB* NBPNI)-CHFTIH*UP (KB)-CBFTIk*LENS NBPNI)-CHFTIH*UP (KB* EHCB 08 BU BETOBN PI-3.141E52652E358SDC C18C*KIND NT-N ABO(I)-0 .DO DO 130 J-1.HDO (-EAX) P •IiB*DEX BEX=C(I,J)*BA* PSI-PSI+BEX *CONST DBUP-SU3*CHARUF(1) DENSUP-DENSOP+DBUP sbb e3 J) 1 TO GO DO VXCB.NE. (I IP DENS=DEJSUP*2. JL (KB) IS TUB SPHERICAL EESSEL PONCTION CF OBDEB L. OBDEB CF PONCTION EESSEL XF K-0. IS S USED IS SPHERICAL BOLE (KB) TUB JL BODE* 5 POINTS BEAL*B(A-F,B,C-Z) 2HPL1CIT GO TC TC 160 GO 150J=1,N DO 88OB V 1 (KB)-CBFTIH*DEN£*B2 (KB)-CBFTIH*DEN£*B2 1 V 88OB (KH*2*NB?N1)-CBFTIN*D0 8U0B METHOD. INTEGRATION 07 FILON B EY *J1(KB)*B*E F(9) INTEGBATE (1) LON ,BFI 808(1) ,8 TPB£LS,TPEB2,TPBB4 1) ( ABO COMPLEX*16 DIMENSION 20 TO GC (NZEBO.NE.O) IF 1-0.DO AB 10 DO X-1,NT IF (2 C A US. EQ.O) GO TO 170 TO EQ.O) GO US. A C (2 IF IF(BA-LE.O. 0) GO TC TC 0) 140 GO IF(BA-LE.O. TO 130 GO IP (EAA.GT.25.D0) IP (IGAUS. NE.0) IP (IGAUS. NE.O) POE. (I XF tWRITE (7,80 ) tWRITE X,Y,Z, DENSOF,8T(KB) CCF,DEL0,DEL 1,DEL2) CCF,DEL0,DEL ED,AB1) CKl C WRITE(6,90 ) WRITE(6,90 B2,X,Y,Z,DENSOP,NT(KB) C 130 COBTX HUE HUE COBTX 130 160 160 150 CONTINUE 150 190 CONT1NDE CONT1NDE 190 170 CONTINUE 170 CONTINUE (41X,F20.5,I5) P08HAT 180 140 IF (IBB.NE.O) GO TO 160 TO GO 140 (IBB.NE.O) IF 183 CALL VXCBS1(DENSOP,DEESOP,DENS,CX,OP,DN,E1, D2,D3,04,DO,CA,CB,CE, D2,D3,04,DO,CA,CB,CE, VXCBS1(DENSOP,DEESOP,DENS,CX,OP,DN,E1, 183 CALL 187 CONTINOE uuuuuu 109 DSN=PHDAGA.DI01A.FCCr 10 CONTINUE 20 CONTINUE IF (II K. IT. 1. D-5) GO TO 120 TU«BK*CRIILK IB2*IH*TH IF(TU. IT. 1.D-4) GO TC 30 SIN1*DSIN (TBJ COS1*DCOS(TH) S1N2*2.D0*SIN1*CCS1 ACPHA=(1-D0*SIN2/(2.D0*TH)-2.D0*SIN1*SIN1/IH2) /TH BETA*2.D0*(1+CCSt*COS1-SIN2/1B)/TH2 AGAHAS4. DO* (SIH1/XH-C0S1)/IH2 GO TO 40 30 CONTI HUE IH3*TB2*TH TH4*IH2*TH2 ALPH A* 2. D0*TH3* (1.DO/45.C0-TH2/315.D0 + TH4/4725.D0) BETA*2.D0* (1.DO/3.DO*Th2/15.CO-2.D0*IH4/105.DO♦TH4*TH2/567.D0) AGAflA*4. DO/3.DO-2.DO*TH2/15.D0+TN4/2 10. D0-IH4*TH2/ J1340. CO 40 CO HU NOE IF (NDlfiC. EQ. 1) GC TO 100 CO 90 I-1.NFPHT £A MBK*BFII.ON(l) CEO31.DO/(S AK+ BAN) CB 1*3. CQ/(BK*BAK) CB2=DSIN (BAN) /BN CB3*DCOS(RAN)/FK SPEEiS=CB2*BFILON(I) SPED2*CH1*CB2—5FEELS-3.D0*CR3/BK SPBH4xSPEEDS IF(1.NE.1.ANJ.I.BE.NBPNT) GO TO 50 APBELS*CR3*B?110N(1) IPBEDS* APBELS IPBH23DCBPLX(CE 1*Ct3-AP3EiS,-3.D0*C£2/BK) IPBH4=1PEELS 50 DO 90 HD31,NDIRC IF(MD.iJ.I) GO TO 60 CL*DFLCAT(2*ND) C1*2.D0*C1-1.D0 C3*C 1-2. DO C5=C3-2. DO C6*4. D0»Cl-6.DO 5PBEDS- (C3*C1*CB0-C6/C5) • SPBfl2-Cl/C5*SPB«4 SPBH4*SPEH2 SPBH2*SPBELS IF(I.HE.1.AND.I.NE.NRPNT) GO TO 60 IPBE1S=(C3*C1*CB0-C6/C5)*TFEH2-C1/CS*IP£H4 TPBflU=TPEB2 TPEH2*TPEELS 60 COBTIBOE IF ( (1/2) *2. EQ. 1) SPBEl.S=SPBEDS*AGAnA IF ((2/2) *2.HE. I) SPB£1>£XSPBE1S*B2TA IF (I.BE. 1.AND. I. BE. HBPBT) GC TO 70 £Ei.S*TPBELS* (1.D0, 1.D0) IF (I.EQ. 1) SP3ELS«SP8EI.S/2.D0*AI.PHA*BE1S IF(I.EC* HE PHI)SPEEIS-SFBILS/2.D0-ALPHA*EELS 70 CONTINUE SPEELSbSPBELS*DBFICN 110 )/RK DSN-PBBAGA.DIOLA.FCOF 0 ABO(KD)-ABO(KD)♦BHOB(IND) *SPBELS *SPBELS I»D=(ND-1)*NBPNT*I ABO(KD)-ABO(KD)♦BHOB(IND) KU=KO«AOIRC SPBELS-DSIN(PA R)*BFI10M1 SPBELS-DSIN(PA KD-ND 00 80 K-1,KIBD BETURN K= 00 110 1-1fNRPNT 00 110 1-1fNRPNT BAK*BK*RF1L0N(I) IF ( (1/2) *2. HE. 1) (1/2)*2.EQ.1) 1) ( IF SPB£L£-£PB£1S*BETA SPBELS-SPEEDS*AGAflA HE. *2. (1/2) ( IF (II) *SPBELS SPEELS-5PB£L5*DBF1LN *RBOB (K) -ABO (A) ABO BETUfiN IND-1ND* NDIH NDIH IND-1ND* (1J/BK AK)*RFILON(I)/BK (I.EQ. 1) IF SPBELS«SPBEIS/2.DO*A1PHA*CCOS(RAK)*RFILOH SPEEOS-SPEELS/2.DO-ALFHA*DCOS(R (I.EQ.NBFN?) IF DO 110 It*1(KIND 11-11*NDIR 11-11*NDIR 00 1-1,NRPAT,4 130 (II)*CC1 *RHOB AR1-AB1 (II)*C0 *F.BOB -«£0 DSN-PBBAGA. CIOIl.fCOF ISO CO till NOE BETUBH END //LKED.SISL1B DO DSN=D1103.CALLAHAY.BVCBP1IB,DISP-SHH / / DO DSN=D1103.CALLAMAY.BNDPKG.SUBIIB.COHPI,D1SP*SHB / / DD DSN=SIS1. VFCRTIIE, EI5P=ShR / / DC DSN=STS2.F0631IB,D1SP*SBB / / DD DSN*SXS2.SSP. LIB.DIS F=SHH / / DD DSN*S*S2.PL07.I.IB,DISP*SHB //GO.F301F001 DD ONiT=3380,VCL*SIB=USEB77,C1SP= (BEN, CATLG), / / SPACE* (TRK, (10,10) ,HL5E) , / / DCS* (BLKSIZE* 06404,BECFH = VBS,LHECI*64) , / / DSNAHE-FUBAGA. CICLA.FEI5P4.ECC.VK1 //CO.PT01F002 DD UNIT*33BO,VOL*SEB*OSEB77#DI SP* (NEK,CATLG) , / / SPACE*(TBK, (10,10),BLSE) . / / DCB* (BLKSIZB*02804, RECFH*VES,LBECL*28) , / / DSNAHE*PHBAGA.DICLA.FEr5P4.BCC.VK2 //•GO.FT02FO01 DD UNIT*3380,VOL*SIB=OSEB77, DISP-(NEW,CAILG), //* SPACE* (T3K, (10,10),RLSE) , //* DCB* (BLKSXZE*064C4,RECFfl*VBS,LBECL*64), //* D5NAME*FHBAGA.DICEA.FEE5P4.ECC.VKJ //•GO.FT02F002 DD UNIT*3380,?OL«SEB*DSEB77,DISP*(NEK.CATLG), / / • SPACE*(TBK,(10,10),B1SE) , / / • DCS* (B1KSIZE= 02B04,BECI(1*VES,LEECL*28) , / / • DSNAHE-PhBAGA.DXOLA.FEF5P4.BCC.VK4 //GO.FllOF001 DD UNIT*33BO,7CL=SEB*USEB77,LISP*(NEB,CATLG), / / SPACE* (TBK, (3C, 10),ELSE) , / / DCB*(BLKS1Z2* 6C00,BECFH = VBS,LHECL*4 0) , / / DSN AKE* PH BAG A.CICLA.FEF5P4.BCC-9KL //•GO.FT11F001 DD UNIT-3380,VOL*SLE*05EB77,DISP*(NEB,CATLG), / / • SPACE* (IKK, (30, 10) , RISE) , //* DCB* (BLKS1ZE=6000,RECFH*V ES,LBECL* 40), / / • DSNARE*PdOAGA.LIOLA.FEF5P4.BCC.NKL //GO.STSIN DC • 5.4057 1.0 2 80CO 2000 400 25 2 3 1 1 0 24 201 20 1 101 0 2 1 0 IEON 4 14 1.0C 1.00 1.0C 1.00 1.00 1.00 0.50 0. 50 257539.000000 0.00 0290 -O.C00090 0.000030 -0. 00 0 0 10 38636.900000 0.002260 -0.000680 0.000250 -0. 00 0050 8891.44C000 C.011520 -0.C03540 0.001310 -0. 000270 2544.010000 0.045660 -0.014150 0.005290 -0. 0011 20 844.777000 C. 140350 -0.046880 0.017380 -0. 003660 312.5270C0 0.314200 -0.117530 0.095270 -0. 009630 125.593000 0.408780 -0.219620 0.085050 -0. 017940 53.498700 0.211630 -0.126610 0.058570 -0. 0129 40 17.715100 0.017650 0.517710 -0.304220 0. 06 8870 7.376770 -C.OC2920 0.604690 -0.495030 0. 114380 2.018470 0.000860 0.054560 0.671410 -0. 192220 0.779935 >0.000410 -0.C095&0 0.593010 -0. 30 3710 0. 114220 0.000110 0.002290 C.011490 0. 579850 0.041889 -C.C0C050 -0.C01060 -0.003120 0. 558820 I BO N 2 9 3.00 3.00 3.00 3.00 1678.40C300 0.002490 -C.CC0900 396.392000 0.020150 -0.007350 126.588000 C.091990 -0.034720 49.115800 0.259910 -0.101890 20.503500 0.42E870 -C .183100 SSN-PHBAGA DIOLJk.FCOF 6.967120 0.326910 1. 104200 3.682490 0.0627 50 i. 321950 1. 521750 -0.001610 I. 569250 0 .5926B4 0.001220 i. 268630 ICtOH 1 5 4.6 2.4 41.452600 0.025110 11.540300 0.136260 3.865430 0.353230 1.3236 00 0.466670 0.416660 0.34 39 50 0 4 0 1 0 1 1 100 0 6001 1 0 100 APPENDIX A.2: New Version of PROGRAM SCF2 DSN-PH3AGA.DIOLA.SCF //PHBAGAOI JCn (1103,602«»5,010,3) , • PH.DIOLA',HSGCLAS5=S, / / N0T1FYKP HBAGA / • AF TE R PHBAGAOO /•JOBPARM SHIFT=N / * ROUTE PRINT PHYSICS / / EJCEC FORTHCLG,PARF.FCRT=• NOSOURCE,LANGLVL(6 6 ).O P T (2) • , // PARH.LKED=NO XREF,REGION=1200K.TIBE=999 //FORT.SYSIN DD * C c c PROGRAH SCF2 c THE FIFTH PART (FERRCMAGNET) OF c C A GENERAI PROGRAM TO CALCULATE SELF-CONSISTANT ENERGY BANOS C c USING THE MODIFIED TIGHT BINDING OR LCGC METHOD Cr* c BT c c C .S . BANG AND J . CALLAWAY c c c TOTAL ENERGY FEATURES c BY c J . CALLAWAY,X. ZOU, AND D. BAGAYOKC c c c DEPARTMENT OF PHYSICS C c LOUISIANA STATE UNIVERSITY c c BATON ROUGE LOUISIANA 70B03 C C C c c ROUTINES USED IN SCF2 t c DIAGHS (FROM PROGRAM BND) c DIGEN (FROM PROGRAM BND) c DBFSD FROM IBM S .S . P. (NOT INCLUDED IN THIS PACKAGE) c DHTDS FROM IBM S .S .P . (NOT INCLUDED IN THIS PACKAGE) c FEBMIE (FROM PROGRAM SCF1) c GBZPT (FRCH FFCGBAH FCOF} c GPERMK (FROM PROGRAM TCOF) c PHTHAT (FROM PROGRAM BND) c SO (10V K (FROM PROGRAM SCF1) c TDBZPT (FROM PROGRAM SCF 1) c TDVL (FROM PROuHAM SCF1) c TELECT (FROM PROGRAM FCOF) c c c COMMON BLCCKS OCCURING IN SCF2 c c ROUTINES IN VHICH TEEY ABE USED c c LCA SCF2 c LCE SCF2 114 KITH K**2.LE.4*K2BAXJI. KITH R .L .V . DSN=PHEAGA. EIOLA.SCF DSN=PHEAGA. CHANNELS USED IN SCF2 USED CHANNELS T (NBFERn,NE) , (NBFERn,NE) PO INDEPENDENT INDEPENDENT OF OF PUT/OUT N3) N3) , FEN N (OUTPUT CHANNEL FT03F001 CF PBOGRAfl INVSIJ) PBOGRAfl CF FT03F001 CHANNEL (OUTPUT VARIABLES) PRECISION (DOUBLE (OUTPUT CHANNEL FT10F001 OF PBOGRAF. CCBBH) PBOGRAF. OF FT10F001 CHANNEL (OUTPUT (OUTPDT CHANNEL F1C1F001 CF PEOGBAK FCOF) PEOGBAK CF F1C1F001 CHANNEL (OUTPDT VASIAELES) PRECISION (SINGLE OVERLAP MATRICES. OVERLAP OF THE POTENTIAL. THE OF 1 FiHOUP.RHCEN (KKFTBM,NBF ERR,NKEZPT), BUCDN(NKFTRN,NEFEbM,NB) , EUP (NBFERH, (NBFERH, EUP , BUCDN(NKFTRN,NEFEbM,NB) ERR,NKEZPT), (KKFTBM,NBF NBTRI=NB»(NB*l)/2, DIflFNSICN OF 5IJ,HUP,HDN,OV 5IJ,HUP,HDN,OV OF DIflFNSICN NBTRI=NB»(NB*l)/2, NKFT=NU.1BER NKFT=NU.1BER FT02F001 ENERGIES AND NAVE FUNCTIONS OF THE CORE AND BANC STATES. BANC AND CORE THE OF FUNCTIONS NAVE AND ENERGIES FT02F001 NKBZTT, DICEN5I0N CF REX,KIY,KEZ,fT CF REX,KIY,KEZ,fT DICEN5I0N NKBZTT, FT06F001 LINE PRINTER. LINE FT06F001 COEFFICIENTS FOURIER THE TO CORRECTION SELF-CONSISTENT 19F001 FT PT01F001 GENERALIZED OVERLAP MATRIX. OVERLAP GENERALIZED PT01F001 NKPT, D1HESNION OF KSX,KSY,KSZ,KNB,ALAHUP,ALANDN,A LAMPA LAMPA KSX,KSY,KSZ,KNB,ALAHUP,ALANDN,A OF D1HESNION NKPT, DIBENSICIN RESTRICTIONS. DIBENSICIN FI0NFQ01 THE NCN-SELF-CONSIST ENT COULOHE, EXCHANGE, KINETIC AND AND KINETIC EXCHANGE, COULOHE, ENT NCN-SELF-CONSIST THE FI0NFQ01 POTENTIAL THE OF READER. COEFFICIENTS CARD FT05F0Q1 FOURIER ELF-CCNS1STENT NOK-S FT08FO01 RHOUP NKDZPT) OF THE B.Z. THE OF 1 NKP(I,I,I) , I.GT.2*S2BT(K2HAXM)* CALCULATE THE SELF-CONSISTENT POTENTIAL AND ENfiERGT BANDS FOR FOR BANDS ENfiERGT AND POTENTIAL SELF-CONSISTENT THE CALCULATE FT 1OF001 ENERGIES AND NAVE FUNCTIONS OF ALL STATES. OF ALL PUNCHER CARD FT07F001 FUNCTIONS NAVE AND ENERGIES 1OF001 FT SPACE SPACE 5CF2 FUP(NBFEBM, FUP(NBFEBM, BEAL* I (JUUUUUUUUUUUUUUUUUUUUUUOWUUUUUUUUUUUUUUUUUUUUUUOUUUU ZETOTL=2 ALPHA*2/3 ,F01 POTENTIAL THE AND IETOTL*1 VBHC FOB , CALCULATION POTENTIAL (1ETCTL.EQ.O) IF THE FUNCTICMANDC OF THE CALCULATION THF 10TAL SAME IS C AS THE FUNCTION THE OP TOTAL SCF OF 1 NKPRT.GE.NUMBER TO APPEAR FIRST.C OF PERBUTED WITHC K**2.LE.K2HAXM. E.L.V. 1DCDB*9 FCC K2 IDCUB*2, BCC 1DCUB*1, flAXH=MAXlH’JN SC C HASNITODEC (K*ACONST/2/PI) OF ENERGY C THE NEB HAMILTONIAN. NKPICL*NOBEER CF NKPTRN=NUMBER CORRECTIONSC C OF NFH TO NBCORE=NDMBER DENSITY(K) CV INCLUDED (K) CONSIDERED WHEN C EACH OF CONSTRUCTING THE IS CCRE ITERATION. STATES FOB EACH KBZD1V*DIVISION SPIN. DIRECTION ALONG C (1,0,0) THE IN B.Z. ENERGY C NBIEPH=NUBBE? OF BAND C STATES FOR EACH SPIN. NB,DIMENSION OF THE HAMILTONIAN C AND OVERLAP ABE NS *NB SSSSS&CSSSCCSSXIKBX£XZIZXtlZItIZSNZXXZSKXK3£X££ XCZTT^SXSCUSKXB IF (KBZHAF.NE.O) POINTS BELONGING C TO THE DOUBLE DIVISION APE SORTED C n n n n n n o n 0 FORMAT20 (2015) 0 FORMAT10 (6F12.8) E , (HDN(1),BHODN(1,1,1)) , E , (HOP (1),BHC0F(1,1,1) ) 0 C O W D P O NA ( 9 0 0 ), I . A M P A ( 9 0 0 ) , D N D P P A ( 9 0 0, ) 2.CXUPOD(90) ,DS0P (90) 6ALABUD A1ARDU(900) (900), ,DNDPUD ),DNDPDU(900) 0 0 9 ( 6D8DPUP(3C0) WT(91), 1,N B C O R E , N B F E R M , N K P T f i N , N K P I C L , ! 2 T C T I 1 , C X D N O D ( 9 0 ) ,D B D N( 9 0 ) . V X D N O ( 9 0 ) , C X D X N W ( 9 0 ) 1 C X U P N U( 9 0 ) , D K U F 0( 9 0 ), D K U P O D ( 9 0 ) , D K D P N R ( 9 0,D ) C O K U P ( 9 0 ), D V A K U P ( 9 0 ) IF (IDCUB.E2. AKGBZ*A1) A K F B Z = KBBZ/2.D0 A K 9/D F L O A T O( K B B E G Z A D * I A V C ) O N S T * * 3 / D F L O A T ( I D C D 3 ) A K R 2 = AA K 9 K * A K R * 2 R . D 0 * P I / A C O N S T DIMENSION VKO (90),BUP0(90) ,BDN0(90) ,VXUP0(90) ,CVKOD (90),CVKNH(90) , ACOKST=LATTICE CONSTANTPI*3.191592653589793 U.) A. (IN COBMON/5PACE/ FUP,FDN C O B B O K / L C A / H U P , 0 V READ( IDCUE.K2MAIM,MB,KBZDIV,KBZHAF,BKPRT ) 0 2 , 5 E L E C T * N U M B EOFEL R E C T R OPE NAT S R O B EQUIVALENCE(PUPC (1)) O IJ ,S (1,1) H H O N / L C B / B H O D N D I M E N S I ODK N U N O ( 9 0 ) , D K D N C D ( 9 0 ) , D K D N N B ( 9 0 ) , D C O K D N ( 9 0 ) , D V A K 0 H ( 9 0DIMENSION ) FDD(15,91),EON(15,91),BHCBN(90,15,91),ALAHDN (900), READ(5,10)ACONST,ALPHA,ELECT A L P H A * T HEX E C H A N GPA E R A M E T E R THE FOIECUING ARRAIE5 BUST BE DIMENSIONED DIMENSION AT HOP LEAST NKPTFN. (0996) ,0V(0996) ,XUP(93,93) ,GXUP{93) DIMENSIOND FUP I M E N ,EUP(15,91) S IFS O (15,91),HHOUP(90,15,91) N Q ,ALAHUP D( 3 0 0 0 ) (900), DIMENSION HDD(0996),XDN(93,93),GXDN(93) DSN*PHBAGA.DIOLA.SCF **2 M BfiX 115 n o o n n o o n 5 CONTINUE35 0 CONTINUE 30 E B E AKS D , ( G 9 ) C O U L , G E BGGALU,GCU,GDD,KSX A D , G H , G V U P , G H D N , G B P A(J) , ,KSY G B U P (J) , KNB ,KSZ(J) G B , C N , G E (0) U D , G B D U , tGDU,CDD,KSX(J) ,KSY(J) , KSZ (J) ,KNB(J) IF(IETCTL. EQ. 1) IF IF (J • GT • NKPTBN) GO TO 30 NKP(KK.JJ,II)=J NKP(JJ,II,KK)-J JJ-KSY (J)♦1 D N D P U D ( J ) - G BD D D N D P U P ( J ) - G B U P C O U L OAN N F , DEX C H A N GPO E T E N T I A L . KSQU (J)-KS NKP(KK,11, JJ) -J NKP(JJ,KK,II)-J B U C P D U ( J ) - G B D U DKLPPA(J)IF -G8?A (IETOTL.NE.1) GO TO 35 A L A B D U < . 1 ) « G A DA D L AA H L A N P U A P (IF J (KS.GT.***KZNAXH) ) (J) - G - A G P A A ' J P GO TO 50 J-J C O E P 1 - C O E F O / O H E G A G E A E R A TEQ E U A L LSP Y A AHALF-ACONS1/2.DOIP C(IDC03.NE.1) EFC D I NAN T S TH ECC E B N IOF E TH S TE E T R A H E D R O N S 1I-KSX (J)+1 ALAJIUD(J) -GAUD B E A DKS ( B , G ) C O U L , G E H A D , G E X C H , G E X U P , G E X D N . G A P A , GBEAD A U P , THE G IN A NON-SELF-CCNS1STINT D N ,G A O D ,CALL GPESHK(KPEBHX, FODE1EB COEFFICIENTS KPEBHY,KPERBZ,KSQD,NKPET,IDCUE,K2RAXN,1,0) THE OF C O E F K — 1.D O /C ( O O H E E G F A O * - 5 OC 0 S f A H l E ) LGB G L A Z P * T ( ( K A i K l X B , B K Z E / Y P I , ) K* B * 3 Z , H T , N K B Z P T , I D C O B , K L , IP S (1DCDB. U H i i ,EQ. 1 AHALF-ACONST , 0 , K B1) Z H CODK1— A P ) 16.D0-PI*P1/(3.DO*OBEGA) ONETHC-1.DO/3.DO NKF(II,KK, JJJ-J »KP(II,JJ,KK)-J KK-K5Z (J)+1 DM EPDN (J) -G3DN ALAHCN (J) -GADN CALL TDBZPT(IPT,NKP,KL,IPTCL,KBX,KBY, KBZ, NKBZPT, IDCOB) 1/M8THIN OP THE INDEPENDENT ER1LL1CU1N ZONE 0-0 N C S T - N BN Z N 3 D + 1 E N D = N B S T * N D P IB H - 1 K L - K B Z D I V + 1 R 0 - ( G H E GNBTBI-NB* A + 3 . D O / (NB*1)/2 ( < 4 . C O * P I ) ) * * O N E T H E M B N O - N B E E R H NBSI-N3-NBT N B T C L - N B C O R E + N B F E R H +1 OL *1 D S N - P H B I G A . ciola . scf 116 117 D5N=PBEAGA.C10LA.SCF DKUPO(J)*GDU DKDNO(J)= GDD IF(KS.EQ.O) GO TO 40 AA=ANRS*DFLOAT (KS) AA=8.D0»PI/AA VXDNO(J) -GEXDN VXUPO(J)=GEXUP IP(IETOTL. EQ. 1) BOPO (J)*GiUP IF (IETOTL. EQ. 1) RDNO(J)*GHDN VKO(J)=GCODL+ AA*ELECT/CBEGA GO TO 30 40 TXUP 0 (J)*GEXUP VXDNO (J)=GEX ON IF (IETOTL.EQ.1) VUPO (J)=G«UP IF (IETOTL. EQ. 1) SDNQ (J) *GNDN VKO(J)*UCOUL GO TO 30 50 NKPT=J-1 BR1TE (6,60 ) ACCRST,ALPHA,ELECT,NKPT,NKPTPN,NKPRT,IDCUB,KBZDIV, £KB2HAF,NS,IPTOL,NKPICL 60 FORMAT (IX,'LATTICE CONST*' , F10.5,2X,• EXCH PARA*',F10.5,2X, 1'E1ECT N0=*, PB.2,2X,*NRPT*15,2X,' K STAR*',15,2X,•K NO*',15,//, 2 IX,' ATOMS/CO3E=',1 5 ,2X, *B.2.DIV-',1 5 ,2X,•KEZHAF*15,2X,'3D N0=', 315,2X,'TEDRONS NO*',1 5,2X,•INCLUDED M *,I5,//) DO 70 J*1,NKPET 1I*KPEPMX(J) JJ=KPERNX(J) KK=K PERMI{J) 11 = 1 ASS (II) 4 1 JJ=llBS(JJ)4l KK=1ABS(KK)41 70 KASIG? (J) -NKP (II,OJ,KK) REBIND 8 IF (IETOTL. EQ. 1) EBEHIND 3 DO 60 1 RLV*1,KRPTRN DKUPCD (IRLT)*0.DO DK DNOD(IRLV) *0.D0 CV ROD(IHLV)*0.CO CXDPCD(1RLV)=0.DO CXCNOD (XRLV)*0.DO 60 CONTINUE C C FACTOO, FACTO, AND FACT ABE PABAHETERS 05ED TO MODIFY THE POTENTIAL C IN ORDER TO SPEED UP THE ITERATIVE PROCEDURE. C FACTOO IS THE FACTO USED IN ITERATION ITEROD. C FACTO H0D1F1S5 THE POTENTIAL OBTAINED IN ITERATION ITEROD BEFORE C CONSTRUCTING THE NEU HA 91LTCN1A N FOR ITERATION ITE60D 41. C FACT MODIFIES THE NEV POTENTIAL OBTAINED IN ITERATIONS ITEFOD+1 TO C ITERNB. C PERFOBH ITERATION NOHBER IT If CD*1 1C IIEfiNN. C IF (NPABT. EQ.O) PERFORM THE CCKPLETE CALCULATION. C IF (NPART.GT.0) ENERGY BAND HAS ALREADY BEEN CALCULATED C IF(NPART.EQ.1) CALCULATE DENSITY (K) AND EXCH (K) C IF (NPART.EQ. 2) CALCULATE EXCU(K) ONLY. C IF (NPART.EQ.3) CALCULATE DENSITY (K) ONLY. C IF (NPART.LT.O.AND.NKBZST.EQ.0) B.Z. POINTS NKDZST AND KKBZED ARE C DETERMINED AUTOMATICALLY. 118 DSN=PHBAGA. C10LA.SCF DSN=PHBAGA. IF (NPABT.EU.~3) CALCULATE BANDS FfiOR NKBZST TO NKBZED , DENSITY, TO NKBZED FfiOR NKBZST BANDS COEFFICIE . FOUEIEE (K) THEIB IF CALCULATE AND (NPABT.EU.~3) EXCH DENSITY AND CHARGE THE IF(IPUN.NE.O) PUNCH IF (NPAPT.EQ.O) 160 TO GC (NPAPT.EQ.O) IF 120 11=1,N3TOL DO IF (NPABT.EQ.-1) CALCULATE BANDS INCH NKBZST TO NK3ZED ONLY. TO NK3ZED INCH NKBZST BANDS CALCULATE (NPABT.EQ.-1) IF FT10F001 PEESISION) (IN SINGLE CHANNEL IN OUTPUT SAVED TEST THE NUJiBEF OF B.Z. POINTS AT NH1CB ENFBGIES AND NAVE NAVE AND 130 TO GO ENFBGIES (NPABT.LT.0) IF NH1CB AT B.Z. OF POINTS NUJiBEF THE TEST NKEZFT , 1 120 NB= DO EDN (II,NN) =ENGE =ENGE (II,NN) EDN HPABI=1 HPABI=1 NKBZ£D*NKBZPT NK 3ZST=NH 3ZST=NH NK IF (1I.GT.NBFDBR) GO TO 120 TO GO (1I.GT.NBFDBR) IF 2 HENIND TO 170 GO IF (til ANT. EQ. -2) CALCULATE BANDS FBCH NKBZST TO NKBZED AND DENSITY. AND TO NKBZED FBCH NKBZST BANDS ONLY. -2) EQ. CALCULATE NKBZED ANT. ANE ABE IF (til NKBZ5T*1 STATES AT NB ALL OF BANDS FUNCTIONS NAVE CALCUL1TE (NPABT.EQ.-4) IF AND ENERGIES NE.O) IF(IBDFNL. FUNCTIONS HAVE BENN CALCULATED. BENN HAVE FUNCTIONS (XUP(JJ,II) ,JJ*1,NB) (2) ENGU, BEAD NK BZED=NKBZPT BZED=NKBZPT NK BEHIND 2 BEHIND DO 140 N B= 1 , NKEZPT NKEZPT , 1 B= 140 N DO EU F (I I , N R) s E NU U U NU s E R) N , I (I F EU GO TO 110 TO GO 2 BEVIND EZST=1 NK 240 TO GO IF(ITEBOD.EU.0) POTENTZAL. NEK 220 IDO s 1,ITEBOD BEAD (2,END=1S0) ENG, (XUP (J J, II) , JJ= 1, N E) , ENGD, (XDN (J J, 11) , JJ= 1, NB) NB) 1, JJ= , J,11) (J (XDN ENGD, , E) N 1, JJ= II)J,, (J (XUP ENG, 140 11=1,NBTOL DO (2,END=1S0) BEAD ITERATION AND BODIFY THEH UITH FACTO BEFCBE CONSTBUCTING THE THE CONSTBUCTING BEFCBE FACTO UITH THEH BODIFY (J),J=1, AND ITERATION (J),ECCKDN(J),DVAKDN (DCOKUP(J)(DVAKOP (19,EKD=250) BEAD GO TO TO 170 GO PBEVIO'JS IN THE , (J) SELI-CONSISTANCV TC DOE , (J) DKDNNH COBBCTIONS THE BEAD DKOPNH , (J) ,CXDNNH (J) ,CXUPNN (J) (CVKNN 60) END*? 19, BEAD( TC 190 GO (1.NE.ITEBOD-2) IF CNKBZST,NKBZED,I FUN,IBDFNL FUN,IBDFNL CNKBZST,NKBZED,I £ £ , ENGD,(XDN(JJ,1I),JJ=1,NB) GNKPTEN) £ J= 1, NKFTEN) 1, J= £ 90 BEAD(5,100,END=780) FACTOO,FACTO,PACT,ITEFOD.ITEBNH,NPART, 90 BEAD(5,100,END=780) 100 FOKNAT (3F10.5, (3F10.5, 1015) 100 FOKNAT 110 110 CONTINUE 120 CONTINUE 120 CONTINUE 130 IF (NKBZST..NE.O) GO TO TO 170 GO (NKBZST..NE.O) 130 IF 150 CONTINUE 150 CONTINUE 140 CONTINUE 140 CONTINUE 160 CONTINUE 160 CONTINUE 170 CONTINUE I I UUUUUUUUU UUUU uuuou n n n n 5 IT£SOD=I-1250 28 FORBAT 0 (1H1,1X,'ITEFATICN=«,13,5X,' IACTOO=• FACTO*•,F6.3, ,5X,• ITERST=ITEROD*1 270 t.3 ,F 0 FOEK300 AT (/,30X,3 BK*, (,3I4,1H) 5X, • CONTINUENfl*'290 ,1 5,/) 3 1 0 FO3 1 0 L K A T( b A 4 ) 23 CONTINUE 0 CONTINUE200 6 ITEROD=I260 2 4 CO 0 N T I N U E CONTINUE220 1 CONTINUE210 8 CONTINUE180 9 CONTINUE190 CSX,•NPART=' J5,//} , IF(NK.IT.NKDZST.AND.ITER.EC.ITEBST) GO TC 410 GO 110 TO IF(NPART.GT.O.AND.ITER-EC-ITEBST) GO TO 490 I F( N P A R T . E Q . - 4 . A N D . N f l . N E .N K B Z S T * 1 . A N D . N B . N E . N K B ZGO E DTO41 ) 0 OV (I J)HDN =£0 (I J) DO IJ*1,NBTRI 320 FACT0=1 DO . C X D P O D ( K N ) = F A C 1 0 * C X U ? N N ( K N ) * (1.D O - F A C T O ) * C X U P O D ( K N ) D O 770 I I TER* I I EEDO 770 ST, I TER Kti RE AD GC,GX, ) GU,GD,GK,GO (4,310 GO 160 TO FACT00-1.D0 NPART=0 GO TO 27 0 C X D N O D ( K N ) = F - \ C T 0 * C X DD N K U N P i O i ( D K ( K N N )♦ ) = (1. F A D C O T - F O * A D K C U T P O N ) » ( * K C N ) X ♦ D ( 1 N . D 0 OCXDNOD(KN)=FACl00*CXDNNb(KN)♦ - F D A C ( T K 0 ) N * D ) K U P O D ( K N ) 1.D0-FACICO)*CXDKCD( ( hN) R T b,8 ) ITERST,FACTOO,FACTO,NPART )HRITE ,280 (b H U P ( IJ )= G C * G K * G U * A L P H A READ THE ORIGINAL COULCHB, KINETIC, EXCHANGE, AND OVERLAP KATFICE3 FACTO0 .DO 1 - C V K O D ( K N ) = F A C T O * CDO KN=1,NKETBN V K 230 N W ( K N O ) *(1.D - F A C T O ) * C V K O D ( K NC ) X D P OC D V ( K K O N D ) =( K F N A )D C U = T O U F O P A » O C C T D 0 X GO 0 ( U R * P TO C N 210 NlF(l.NE.ITEROD-l) V ) l i ( = K K N N F ) A t i ( ♦ C K T ( N O 1 J . O *(1. D D * O O D - - K F F A U A C P T CDKDNOD(J)*DKDNNN(J) N O T H O O( ) K * O N C ) | V * ♦ K C ( O X D U1( P . K 0 DC i 0 ) N - ( F X K A ) C N D T ) 0 H 0 ) * O D D K U( J P ) * O C D X ( D K N N N V ) ( J ) O 6 NK*1,NKBZEDNRITE KBX (6,300) DO (NK) 460 , KET(NB) , KBZ (NB) ,N9 F A C T O * 1.D O NPART=2 D K D N O D ( K N ) = F A C T O * D K D N K K ( K N ) ♦ ( 1 . D C - F A C T 0 ) * D K D N O D ( K N ) D K D N O D ( K N ) = DKN F A O20 0 C = T 0 0 1 * D , K N D K N P N T R i N l ( K N ) ♦( 1 . D O - F AD C K T U O P O O ) D * D ( , J K ) = DC D N X K U C J P P D O N ( D H (J) K ( ( J JCVKOD(J)=CVKNM N ) ) = ) C X J P N N ( J ) DO 180 =GC*Gi.'* = J 1,N K P T B A CD*ALPHA DSN=PHBAGA.DI01A.SCF 119 non r> n n n 34 CONTINUE 0 B BRITE(IO)3B0 GEIIP.GXUP 44 CONTINUE 0 2 CONTINUE320 6 CONTINUE360 350 400 7 GXUP(JJ)370 *XUP(JJ,XI) 3 BEAD(1)430 KS.KT.SIJ 3 CONTINUE333 2 BEAD420 GC,GX.GU.GD,GK,GO ) DO 1J-1.NBIBI(4,310 410 420 GXDN390 (JJ) =IDN (J J,II) CONTINUE , HDN (XDN, (II) ,JJ*1,N3) (JJ,II) 6 I=II-KB5T*1 IF(IBDINL.E^.O) GO TO 460 GKOP“ HUP(II) IF (IfiDFNL.EC>. GO TO 46 0) E D N (I ,K ( I ) * H D N (II) DO I I = 360 NB5T,NB GO TOIF(NPABT.EJ.-4) 46 0 GO TC 460 BEAD KS.KT,SIJ (?) (1TEB.EQ.1)IF GO TO 350 E D N , (II N CALL DIAGHS (I1DN.OV XDN,N , E, 2, IP'JN,0 0,KX,KY,KZ, , 0, 0, NBST, NBEND) HOP (I J) = H0P (I J) ♦AA*SIJ (IJ) DO IBLV=1»KNFTBH 340 CALCULATE THE BABIETONIAN CETAINED FP03 LAST ITERATION BY THE DO KSP= 450 1 ,2 BEAD ENGU, (2) (XTJP(JJ,1) , JJ*1,NB),ENGE, (XDN 1) (JJ, ,JJ=1,NE) E0P(I,NIt)=BUP(II) C A LDX L A G H S ( 1 1 3 F , 0 V, X U P , N B , 1 ,IP U N ,0 , 1 , K I, K Y , K Z , N 0,0, B S T , N B E N D ) DO XBLV*1 430 IKFTBN , DO SK1TE HOP(2) , ME)JJ=1, ,(XnP(JJ,II) (II) K Z - K D Z ( N H ) K X = K B X ( N f l ) WRITE DO J 390J*1,NB DO 11*1,NB 400 DO IJ= 1,NB1KI 330 B B = C YA K A O = C D V K ( IIF E 0 (IBLV.GT.NKPICL) L D V ( ) I+ R C L GO V X*C ) TO D 340 X N 0 P C O D ( D I P L ( V I ) C * L V A ) * A L L P P U H A A G E K E B A 1 I Z EOV D A B L ABA F I E 1 C E S . E U P ( I I , N N ) = E N C U DO 3B0 11=1,KB IF (I.G3. NBFETTfl) GO 360 TC D 1 A G O N A L I ZTH EBA E E I L T O NK I Y A N = K B T ( N B ) GELN=HDH (II) HDN (IJ ) = HDN(IJ)+BB*SIJ (IJ) IF (II.GT.NBFJRH) GO TO 440 DO II*1,NBTOL 440 7 JJ*1,NB 370 1) GEDN,(10) GXDN (!) =ENG D 0 DSN*PHBAGA.LIOLA.SCF 120 n n n o n n o n 4 CONTINUE540 AA5 3 0 = A A « A C 2 CONTINUE520 1 CONTINUE510 CONTINUE500 490 IF IF (NBFERH.490 GOEQ.O) 5C0 TC 4B0 IF (NPART.E0.-1-OR.NPART.EQ.-4) STOP CONTINUE4b0 4 7 0 NK4 7 0 B Z S T = N R 5 EL450 AD {10,IN 0=47 GEOP,GXUP 0) ,FNGD,(XBN(JJ,I1),JJ=1,NB) & IF (NPART.GT.1.AND.ITER.EQ.ITERST)IF GC TC 650 DC u=o IF (IN. Lfc.NbFSR K)BB=BB4BCGO TO 550 D C O K U N ( Jf L V ) = D C C K D K (IR L V ) * 1 E * W T ( N . * 1 ) GO TO 290 DO 12=1,11 540 1Du 540 1= 1,NB AA=0.DO DO I RLV= 570 1, NKPTRN IF (11. GOIQ. TO12) 530 D C U K UDO IRLV=1,NKPTRN P 510 ( I R L V ) = 0 . D O INTEGRATIONTHE B.Z. C FOR A L TEE C CORE U L A STATES TFO E U ARE R I TH OITA1NLL E R A N BY S H1STOGRAJ1 F O F .OF HCH A R GDE E N S I T Y AC =1J XUP(I= IJ*1 1,1N) * SI BB=0. DO DO 1N=1,N3TOL 560 READ KS.KT.SIJ (1) BC=BC*2.D0 A C = A C * 2 . D O C V K OK D R E T R OAN D DTHPA E N ST E AEY TAN E S A L Y T I C ATE L T I ' A H ECALL D H I ONEERNIE N T r i O ( D . SU r ,EDN, NEFERfi, KKBZFT,1P1,IPTOL, COEFO, ELE, NSI,3N,FERHI) REWIND 1 GO TO 460 BC = JiDN (II,IN)*SIJ(IJ)*XDN(12,IN) DO 11=1,NDTO 520 REWIND 4 REWIND 10BEHIND 4 READ ENGU, (2) ( ,JJ*1,NB) YUP(JJ,II) DO 11=1,NB 450 DH57 O 0 F = 1 , N K E Z P T LLE=ELICT-2*NBC0RENSPIN=2 REWIND 10 BE WINDEE 2 WIND 1 DEIERRIN E THE IEf.HI ENERGY REWIND 2 UK 'J r (I (IRLV) =Q.D0 L) =DCOKUl:LV) I )* (IJ J L ( 1 R I V )* A A * W T ( N H ) XUE(12,IN) D5 N=PHBAGA.DI0LA.5CF 121 n n n n 5 CONTINUE650 6 CONTINUE660 3 COHTINDE630 FORMAT620 FORMAT610 (415,4E15.8) (415,8F13.6) 5 6CO 0 N T I N U E BHOUP(IRLV,IN,NH)=AA550 64 FORMAT 0 , 1 / / ( X, * ELECT* *, F10. 6, 5X, • fl AGNETCN NO=',F10.6) CONTINUE600 FUP(NK,KET)590 = RHC0P(1RIV,NK,KFT) 500 7 COKTINOS570 6URITE(7,620) IBLV,KSX (IBLV),KSY (I RLV) ,KSZ(IBLV) ,SUBUP,CUfl'JP, C S U R D N , C U B D N 6 S U B U P , C U B U P , D K D N 0 ( I B L V ),SD N , 5 U B D N , C U I 1 D N £,5X, *C0NC*,8X,' COE E',7X,*DENSITY OB',3X,1NEH DEH DN*. ,NENO)6 ,NENO)6 6 S X , ' C O N D * , 9 X , * C O B E * , / } BEHIND 1 IF IF (I PUR. NE.O) D K U PDO N KN=1,NKPTRN 670 W ( K N ) * D C O K U P ( K N )+ D V A K U P ( K N ) - D K U P O ( KB N E ) (D A D C ( 1 O 9 ) K U P( J ) , D V A K U P ( J ). D C O K D N ( J ). D V A K D N( J ) , J = 1 , N K P T R N ) URITE (6,610) IRLV.KSXS O N *(1RLT) 5 U B D , KSY N * C 9 M (IRLT) DD ,KSZ(IRLV) C M C K ,DKUP0(lfiLV) U P ( 1 R .SOP, LC U V B ) O = P * C D C t ) O M K U U P ( I P B L V ) * A A * C O E P K CA1L SUBOVK(EDN,IPT,IPTOL,REX,KIY.KB2,FEN,NKBZPT.SUHDN, COE FI,FERBI DO IBLV=1,NKPTRN 630 FORB BOD2FIED THE SELF-CONSISTENT CORRECTIONS TC IBFRCVE CONVERGENCE. A M A A E G L * C ( D = ( C D O C K O U K P U (1)+ P D E (1)+ V V A K A U P K ( 1 U ) 4 D P C (1)- O D K C O D K N D (1)+ D N V (1)- A D V K A D K N D ( 1 N* ) ) ( O 1 ) M ) * E C G H E A G A DCOKDN (IBLV)=CDBDNC U B D N = E C O K DD N ( V I B A L K VD )* D A N V A * ( IF C A L O T )= E K S U F K B O E P N (IR L V )= S U B U P NK=1.NBN0DO 590 IF (NENO.AA=1. EQ.O) GO DO/KNB(IRIV) TO 600 S 0 K D N = 0 . D O GO TC 560 GO TO 6b0 WRITE (DCJKUP(J),DVAKOP(J). (19) DCOKDN DVAKDN(J),J=1, , ) J ( NKPTRN) 5 U B D N * S U B D N * A A CALI SUMOVK(SUP,IPT,IPTOL,KBX,KBY,KB2,FUP,NKBZPT.SDBUP,C0EF1,FERMI WRIT E AEIC.ABAG (6,69 0) S U P * S U B ’ J P + C U M U P 5 U B U P * 5 T N U P * A A F D K ( N K , K P T ) = E H O D N ( I R L V , N K ,K P T ) S U H U P = 0 . D O BHCDN(IBLV,IN,NB) =3B HI.IT E( 6, 580) DO KP1=1, NKBZPT 590 (NPABT.EQ.-2.0R.NPART.EQ.3) STOP AT (1H1.7X,*KX',3X,*KY',3X, D S N = P H B A G J l . D I 0 1 . i l - S C F * KZ* 3X, , * DENSITY UP*,4X,*NEH DEN OP* 122 DSN=PH EAGA. CIOLA.SCF DKDKKV (KN) =DC0KDN(KN) *DVAKDK (KN) -DKDKO (KN) DKUPNH (KN)=FACT*DKUPNW (KN)* (1.D0-FAC1) *DKUPOD (KN) DKDNNU (K N) = F ACT* CKEfcNtf (Kfc) *(1. 00-FACT) *CKDNOD( KN) 670 CONTINUE C C CALCULATE THE CORRECTION TC THE EXCHANGE P0TBET1A1. C DO 690 KN= 1, H K FT B N SUODP-0.DO SUHDN=0.DO SOf!tiOP=O.DO SUNK DN=0.DO DO 680 J=1,NKPRT II=K5X (KN)-KPi!bHX(J) JJ=KSY (KN) -KPEPHY (J) KK*KSZ (KN) -KPERBZ (J) 11*1AB5(II) *1 JJ=1AES(JJ) *1 KK=1ABS(KK)*1 KS=NKP (II,«JJ,KK) KT*K A51UP(J) SUBUP=5UHUP* ALAKUP(KS) *DKD PNN (KT) ♦ALAHUC (K5) *DKDNNN (KT) SUKDN=5U HD'S* ALA KEN (K5) *DKDNN» (KT)* ALABDU (KS) *DKDPNk (KT) IF (IETOTL. Eg. 1) SUHHUPsSURhUP*DNi)PUP (KS)*DKUFNH (KT)*DSDPUD {KS) £DKDNNfc(KT) IF (IETOTL.E*.1) SUBWDN*SUBWDK+DNDPDN (KS)*DKDNNN(KT)*DNDPDU(KS) CDKOPNN (KT) 680 CONTINUE CXUPKK (KN)=SUHUP CXDNNU (KN) =SUBDN IF (IETOTL. EQ. 1 ) LNUP (KK) = SU Hi OP IF (IETOTL.E... 1) DNDN (KN)-SUBKDK 690 CONTINUE C C CALCOLATE THE CORRECTION TO THE COUICKE PAET CF V(K«0). C AA=0.DO R02=R0*R0 R03=RC*RC2 DO 700 J=2,NKPTRN EK2=AKb2*KS00(J) RK 1=D5QRT (RK2) BK3=RK1*RK2 BKN=RK2*BK2 R0K=RG*DK1 C0= (DKUPNH (J)*DKDNNH (J))*KKB(J) AA*AA*CO/EK1*((3.D0*RO2/RK2-6.D0/RKN)*DSIN(RCK)♦ C (6.D0*RC/FK 3-f03/RK1)*ECCS (ROK)) 700 CONTINUE CTKNU ( 1) *CODK 1 *A A C C PRINT FOURIER COEFFICIENTS OF THE NEH POTENTIAL. C 710 F0BJ1AT{1H1,1X, 'ITERATION*',13,5X,• FACTO**,F6.3,5X,•FACT=•, F6.3 6•NFA Rl=* ,15,//) HR ITE (6,710 ) ITEE,FACTO,FACT,NPART NR1T E(6,720 ) 720 FOKHAT (2 X,' KS' ,2X,' NUCLEAR* ,6X, 1 COUL', NX, * EEL NEH', NX, »DEL OLD nnnnnnnnnnn 0 CONTINUE800 8 CONTINUE780 FO7 6 0 F . H A T ( 1 H 1 ) 7 CONTINOE770 75 POBFI 0 CONTINUE740 ) AT (lXtI2,2I1,12(1X,F9.6) 3 FOHr.AI735 (1X, (IX,12,211,4 F9. 6) , SX , 8 (1X, P9. 6) ) 72 rOB(1kI(52X,, 5 HDP(KS) • ,3X, • DVU P (KS) * , 3X, * DHUP (KS) • , 3 CONTI730 NOE 6 J =1 , K K F T R N ) 6 J *1 , N K P T R N ) G J =1 , N K P T R N ) C UR KSX 1TE )(J),KSX(J) ,KSZ(J) ANOCLE,V , (6, 735G) , VXUPO(J) KO ,CXUPNN (J) ,CVKNK(J) ,CXOPOD(J),CVKOD(J (J) ,TXDNO(J) ,CXDNNH (J) ,CXDNOD(J) GNF.3TE (7,620) J,KSX(J) ,KSX (J) ,KSZ(J) ,CVKNH (J) ,CXUPNH (J) ,CXCNNV (J) 631 G, 2 X, *DEL NEH*,3X,'DEL NEH*) C) ,X*XF,X'E NEHG,4X,*EX0F',4X,'DEL CLt ,• *,4 X *Ell , HI, • EXDN* ... 3UE,SV,ai BS35S 5 3 S B H Z E C C X U S t t Z i a , V S , E U 3 S K 3 C I « 3 U n i Z U » Z X K S . . . . . i - i w KS2(J)*KSX(J)*KSX(J)♦KST DO BOO (J)*KSX(J) ♦ J*1,NKPTRN KSZ (J) • KS Z( J) GO TO 90 OUTPUT AEE FZLE 37 DATA OF SETS HRITE (VXD (37) NOHR(J),CXDN ITNHHRITE E CKDNNB(J),J=1,NKPTRN) (7XUP0 , (K52( ) J ( (17)37) (J) ,KNB(J) (J) ,CX0PhH ,DKDN0(J) (J)HR1TB(17) ,DKDNNH , (KS2DKUPNH VKO(J),KNB ). (J (J).CVKNH(J) (J) (J) , 1, ,J= NKPTRN) , OUTPUT ABE 17 PILE DATA OP SETS IP (ILTOTL.EQ.0) GO TO 610 C X D N O CC ( J X ) - U C X P O D D N ( J N ) - C H X D (J ) P N H( J ) (IETOTL.IP EQ. 1) CVKNU(J)*AA*(DKUPNH(J)+DKDNNH(0)) IF (IETOTL.EQ.1) OOTPOT ABE DATA 27 SETS OP PILE BEHIND 19 PE HIND 2 D K D ND O K D O P ( O J ) D -C ( D J ) V K - D K D K N O U N D P H N (J)(J H ) = ( J ) C V K l l i i(J) HRITE (6,760) HBITE(19) (CVi;OD(J) CXUPOD , (J) .CXDNOD(J) ,DKUPOC(J) ,DKDNOD(J) , IP (IPDN.NE.O) * * ■ = 8 . B D 0 K * P 2 1 = / B CO A K TO 2 K 730IF R (J.EQ.1) 2 * J=1,NKPTRN K 5 Q UDO 740 ( J ) IP(IETOTL.EQ.1.AND.ITEB.EQ.ITEBNH) NBITE(6,725) N T 670 KSX )UNITE (J),KSX (J) AN3CLE,VK0 ,KSZ(J) (6,750 , (J) , CVKNH (J) ,C7KOD(J A N U C L E * - A A * E 1 E C T / 0 N E G A ANUCLE-0.0 , HDPO(J),DSHP(J) , DHOP ,* HON (KS)*,3X,'CHON(KS)• ) / / , V X D N ( K S ) , D Y X D N ( K S ) , D P D N ( K S ) VXUP (KS),DYXUP(KS),DPOP(KS) HUP (KS),DK0P(KS),DENST0P(KS) NBK52(KS), (KS),HDN(KS), DHDN(KS), KS2 (KS) , NB (KS),VE0 (KS) , DTE (KS), KS2 NB (KS) , DENSTDN (KS), (KS), DPDN (KS), (J) DSN=PHBAGA.DIOLA.SCF ,NENO(J) ,EHCN(J) 124 A\\\ VS © Cl C? n Cl © © © © r t if* o o o o o O O o o • to i ■ • 1 • • • « • • w O * (A to to to to to to to to 9 © * M i-9 H 9 MMM 1-1 H 9 * p i (OKLCH cn ia CJ ro r t r t 9 r t O O c © o 9 9 0 0 9 W Z H W H W to O' - J M •o to to © o in o © 0 * n —9 9 9 9 to 9 O rt m to m. to zoot/»tooo H n rt rt II O rtCntdH t r t r t r t to r t 0 to ^ to m H W to to to o r t * • U 9 9 3 m . W Z 9 II z 9 -o 9 UlTTTf 9 ■o H *0 H • Ul O ft I J E t 9 9 to H 9 to U II 9 m to II 9 ► * — 9 9 © 0 © r t © » ► o vs © w W W w rt — ftf — to o © if 9 H 9 m .O II 9 N 9 m . O 3 M H D in © to © 3 © w n © © (A © M K *< «<- • 9 © to Z H to Z ^ to z to to a to W M tn z Z to h rt in m to W to to o © o IW SCZW ^CDCw'^OttM SC « to Z a 9 a r t © it c M « c Z N G W tU to Cu VS rt r t o tu H Vi i z t u h ^ ■-;DHZZtJM z*to9* z a 5; £_ to a ■ » • • • 9 cs (A • o o r t N " w >- N *• M rt 9 * H r t M * m fc h m . M to M W H 0 H *? H H 5 Vf M s w n < 0 w to to -» © Cl 9 —k © P » M H Cl to to © W m ►3 to O Ui H ii H II H 0 H > M H H w o n ft* z 9 r t H Co H r t H to Il r t II UJ h rt 11 ro II r t to O n to II to II rt II I SK M T2Tf o s s o rt II *tl • C * U • t « u* • c *. Ul a * O t o Cl t o * to EE UJ 0 Ui © Ui i* u> >n Kj Up ►3 • M 35rt to w Cl o t9csiocj9© rouiacD toui9© * w rt ro u 0 W 0 w > w •m - U* 3 U • 9 9 H rt rt • n « r t H CD *“* CD H OD w CD M 00 Ul © a p a B r t 0 r t 0 « 0 - Ul CD «1C ® to m m 9 « rt a w o% ^ o o « w © r>* •— © o* 0 9* —’ c © © © © t t © O Ort * o M 03 CO H rt 9 Z H r t 9 10% % 9 t o * * p 9 % * 9 « — « H t o « r t • r t « M •to i« «© x - w * * m •4 Ul 9 n t o to r t 9 < r t 9 to r t 9 to r t W ^ to O W * < • to * < o < i K> ro cn 126 APPENDIX A.3: New Version of PROGRAM DENST DSN=PBBAGA. DIOLA. DEN //PH BAG A 54 JOB (1103, 60245, 004,2) PU.DI 01 A* ,HSGCIASS*S# / / NOTIFY=PHBAGA /•AFTEH FEFIB254 /•ROUTE POINT PHYSICS /•JOBPARH SHIFT=D / / EXEC FOBTHCLG.PARH.FORT-'N050URCE.LANGIVL(66) .OPT(2) •, / / PAFB.LKED= NOX £ IF , t EGIC N=12 0 OF, TIP! 1*999 //FOBT.SYSIN DD • C C FROG PA.1 DENST C THE SIXTH PART OF C C A GENEBAL PBOGEAH 10 CALCULATE SELF-CONSISTANT ENERGY BANDS C C USING THE HODIFIED TIGHT BINDING OB ICGC HETUOD C BY C. S. NANG AND J. CALLAWAY TOTAL ENERGY CALCULATION BY CALLAB AY(X. ZOO,ANt D. BAGAYOKO DEPARTMENT OF PHYSICS LOUISIANA STATE UNIYEBSITY BATON BODGE L001SIANA 70B03 c ROUTINES DSED IN DENST c FEfitllE (PR 7(1 PEOGHAN SCP1) c GBZPT (FROH PROGRAP1 FCOF) c TDBZPT (FBCH FROGRAN SCF1) c TDENST c TDVL (PROP! PROGRAR SCF1) c TELECT (FBCH FEOGRAH FCOF) C c c c INPUT/OOTPUT CHANNELS DSED IK DENST c c FT01F001 ENERGIES AND HAVE FUNCTIONS. c (ODTPUT CHANNEL FTC2F001 CF PROGRAM BND OR c OUTPUT CHANNEL TT10F001 OF FBOGRAN SCF1 OR SCF2) c FT05FO01 CARJ READER. c FT06F001 LIRE PRINTEB. c c c c c CALCULATE THE DENSITY OF STATES BY THE LINEAR ANALYTICAL c TETRAHEDRON METHOD. c ooooono FORMAT333 (1015) n n n o n o o o n o ft n o o n o n n o nnonnnnnnnnnnnn 0 FORMAT10 (2015) CVK 0(4 0) ,BUP0(4 0) ,DVXCUP(40) ,PUP0(40) LES1NGU (1801).ESI NGD( 1801),SH(1801) C . D H D N ( 4 0 ) , D P D N( 4 0 ) , V X C D N 0 ( 4 0 ). U D N O ( 4 0 ), E V YCKS C D N ( 4X(40) 0 ) , E , KS D N 'Y J ( 4 0 ) (40) , KSZ (4 0) , KNB(40) K2 , (4 0) C,NKPTRN C O N S I D E R EEQ D U A L . r s c s s s s s n u B c t s s i f ENERGIESIF(IDISC.EQ.O) ARE READ FROM CARDS. IMPLICIT REAL*8{A-F,H,0-2) IPT ,IPTOL) VUE (4 RE THE IPTCL NUHBFR IS OF TETRAHEDRONS 1/4B»TH IN C DU=, C 1EC0B=2, BCC IDCUB=1, SC TO APPEAR FIRST. DIRENSION CVKNW(40),DHDP(40), DPUDIMENSION P(40) , VCKO EUP(15.506),EDK(15,506) ,GX VYCUP0(40), (4D), INTEGER*^ KX1.KY1,KZ1,FNE OF THE B.Z. READ EMIN,DE,ZERC,ELFCT,Z,ACONST (5,20) KHHAX,KBZDIV,KBZHAF,IDCUB,NSPIN,NBFERH,NBCOHE,NE,ID1SCREAD(5,10) READ(5,333) NDENS,IETOTL CALCULATEIF(IETOTL.EQ.2) THE TOTAL ENERGY AND DSE ALPHA=2/3 POT. REAL*4 (0346) J I S DIMENSION RESTRICTIONS. E L E C T - N U M B EBAOF R N DEL DE=ST£PS E C T R THF ENERGY IN O N PE AT S E O DIVISION. M A RCA E L C U L A T E D . 1NTEGER*2 IPT(X,200 8),KBX (506),KBY (506)»KBZ(506),NP (21,2 1, 2 1), F U P ( N B F E R M . N 3 ) , E UNKBZPT, P ( N D DIMENSION F E R M ,N K B OF KBX,KEY,K3Z Z P T ) ,G X( N 3 ) E O I ENERGIESZERO, IF DIFFERENCE LESS THAN IS ZERO THAN ENERGIES ARE IF(IETOTL.EQ.O) THE FUNCTION THE SAME IS AS THE FUNCTION OF DENST ENERGIESIF(IDISC.NE.O) ARE READN B , D I FBOF M E CHANNEL N S I OTHOF NHA E M FT01F001. I L T O N I AAN N OV D E R L AAR P NB E * N B IF(KBZHAF.NE.O) POINTS BELONGING TC THE DOUBLENEMAX*NUMBER DIVISION ARE SORTED DIVISION OF ENERGY IN AT VHlCli THE DENSITY OF STATESDIMENSION EN(1801) SWDN(1601),TU0P{1301),TVDN(1801) ,SVUP(1801), , 1=NP , KBZDIV (I,1,1) + 1 E M I N = H I H I H UEN N E R GCO Y N S I D E R E D . CALCULATEIF(IETOTL.Ej.1) THE TOTAL ENERGY AND DSE VBH POTENTIAL A B C O RN3F£3M*NUnB£ii' E = N UNSP1N M AND >1 B FOR E CF PABAKAGNET 2 BANDOF R TH CO E STATES R ST E FOB A TFO EACH E ANDEA S R SPIN. C FEEROMAGNET SP H I N . RESPECTIVELY. KBZDIV=DIVISION ALONG THE (1,0,0) IIRECTION B.Z. IN NEKAX, DIMENSION OF EN,SWUP,SVDN,T«UF,TliDN. IF(NDENS.EJ.O) THE DENSITY OF STATES NOT IS CALCULATED s t e s s s s u s c R s c s s s s a s s c s s s r s x v d s s s s s s s c s s s c S E s s DSN=PHBAGA.DICLA. DEN C 1 ICC CCDE*4 (4 3),GX1 (4 3) 127 non oonono noni 1 FORM810 AT (IX,•A TOM/CUBE*•,15,3X,'B.Z.DIV = *,IE,3X,'NE=',I5,3X, 0 CONTINUE70 0 CONTINUE60 CO3 0 N T I N O E 0 CONTINUE80 CONTINUE50 FORK«0 AT (6F11.7,313,I5) FORMAT20 (10F10.5) 'MN'FOSajDs FOSi.T NC=»,I8,3X, jFIO.SjiX.'TD C'EMINs'jFIO.S.aXj'DEs’ 2 '3 .2 .PT=*,15,/1X,' CORE* NB=>,15,//) .PT=*,15,/1X,' ,, .2 '3 ,l5,2X 2 a,l5,2X,'ENAD=, 6N3FEBM,NB «8«SSCCSKStnrSISS89XKXeSXXSS£SCZSC*SS8SBtUSSX«SS SSSISS 3C «8«SSCCSKStnrSISS89XKXeSXXSS£SCZSC*SS8SBtUSSX«SS IF (NSP.EQ.2)IF EDN (J-NBST*1,K)=GENG (IDISC.IF NE.O) GC TO 50 IF(J.LT.NBST.OB.J.GT.NBEND) READ GO TO 60 CENG,GX (1) BE (EDN(1,K) AD ) ,I=1,NBFEHH) (5,90 CALL TD3ZPT(IPT,NP rKDIM,IPTOL,KBX, KEY,KBZ,NKBZPT,IDCU3) IF(NSP. EQ. EUP[0-NBST*1,K)=GENG 1) ED t*= BAND STATES ENERGIES CF MINORITY SPIN. IF(IDCUD.EQ.1) C0NST=CCNST/8.DO B T(,1) IDCUB,KBZD1V,NEBAX,EHIN,DE,IPTCL,NKBZPT,NECORE,NBITE(6,810) K D I H * K B Z D I VBENIND + 1 1 J* DO1 6 , N5P*1,N 0N DO SPIN3 60 KBX(K)*K¥ KX* COMPONENTS X OF K POINTS THE IN B.Z. C A LGB L Z P T ( K 3 X , K E YC , O K N - B C O Z N , CONST=CCNST*2.DO SIF(N5P1N.EQ.1) T * ! . D O CONST*8 DO/DPLCAT . GO TO 80 IF (NSPIN.EQ. GC TO 1) 30 READ (EDP(I,K),I=1,NEFERM),KX,KY,KZ ) (5,40 COMPONENTS ¥K¥* OF POINTS THER IN E. Z. O 0 K*1,NKBZPTDO 70 K B X ( K ) * K X K=1,NKBZPTDO 30 K I = K B Z D 1 V * 1 NBT6I=N8*N(NB*1)/2 B E N DN * B S N T B = S N T B * — N 3 N F B EK B T B O H L - N 1 * 1 O * P N O E * Z F / O E H BC E M G M A E G A * A A A /D F L G A T (IC C U B ) RBZ(K) *KZ KZ= COMPONENTS Z OF POINTSK THE IN B.Z. PI* 3. 1*11592653569793 N B T O L * N 3 C O B E + N B P E F f l EUP*BAND STATES ENERGIES OF MAJORITY SPIN. A t A * I C O N S T * * 3 (IDCUE*K VI, DSN=PHSAGA.DI OLA.DEN BZDIV**3) N K B Z P T , I E C U E , K L , 5 U H N ,0 , 0 , K I Z H A F ) 128 noon n n n o 8 3 4 FG8 3 4 N A A T 1 (/, X , * C V K t i i i(J ) , J = 1 , H K P T R N • ) 5 FORMAT654 IX,'DP (PARAHAG) , (J) / ( CR DPUP(J) (FERROHAG), 0*1, NKPTRN') 5 FORMAT850 IX,'DENS , / ( (PARAHAG)(J) OR DENSUP (FERROHAG), (J) J* 1, NKPTPN*) 4 FOBBAT640 , / 1 ( X, • DVXC FORMAT838 (PAEAHAG) J)( , 1X, / ( CF DVXCUP(J) ' V XCO (FERRO (PARAHAG) J)( RAG), J=1, NKPTRN*) OR VXCUPO (FERROHAG) (J) J=1,NKPTFN«) , FOEH824 ,NKPTRN*) 1 1X,*KHB(J), - , J AT( / CONTINUE818 4 POF.B648 AT (PARAHAG)(/,1X,*DH(J) CR DHUP(J)84 FORBAT 4 (FERROHAG), (PARAHAG) IX,'HO(J) , / ( 0*1, NKPTRN*) CR HUPO (FERFOfiAG) (J) J=1,NKPTRN*) , CONTINUE830 2 FO828 BH 1X,*VK0(J) , AT( / J , = NKPTRN•) 1, FOEH820 AT 1X»* , / ( K 2(0) , J = NKPTRN*) , 1 1 CONTINUE814 CPE PHI) HE IT E (PUPO (6, 858) , 1 (J) NKPTRN) ,J= HRITE (DVXCUP (0,858) (J),J= 1,NKPTRN) HF IT (CVKN.(J)£ (6, 1, 858),J= NKPTRN) AK S2= D0*PI/ACON . 2 ( ST)**2*DPL0AT (AK) IP(NSPIN. EQ. GO 1) TO 818 HE ITE (6, 854) HRITE (DHUP (6,858) (J) 1,NKPTRN) ,J* HRITE(6, 640) HRITE (VXCUPO (6, 856) (J) ,J= 1, NKPTRN) HE ITE HRITE (HUPO(6, 848) (6,858) (J) 1,NKPTRN),J* HRITE (6, 844) HR IT E (6, E34)VCKO J)( = VKO (J) . 8 D0*P1*Z/CHEGA/AKS2 - HRITE (VKO(J),J*1,NKPTRN) (6,856) READ (K2(J),KHB(J)(37) READ , PDNO READ(27) (HDPi>(J) (J) (K2(J) , DFEN ,KNB (27) ,DKUJ (J) ,WDN0(J) (J) ,J=1,NKPTRN) (J) ,PUPO(J) ,J=1,NKPTRN) ,D8DN(J) 1, ,J= NKPTRN) IP (NSPIN. GO TO E£.2} 814 HR ITE(6, 650) C A L C U L A TTH EAV E E R A GCALISI E FERHIE(EUP,EDA, N GEL L E E C T RNBFEEB,NKEZPT,IPT,IPTCL,CONST,ELECT, OEN N E P G T ,AN D , U TH D E US PIN, HR IT E (6, 638) KK=K2DO(J) 8 J-2.KKPTRN30 VCKQ (1)=VKO (1) HRITE (KNB(J), J= ,NKPTRN) (6,1 10) HRITE (K2(J),J*1,NKPTPN) (6,10) BEAD (K2(27) (J),KNB(J) ,V KOREAD (J) , PUPOREAD(17) (VXCUPO (K2(J),KNB(J) (J) ,(17) ,VK0J= 1, (J)NKPTRN) (J),DVXCUP(J) , CVKNfcCOEFFICIENTS ,DPUP(J) OF NRPInN) THE , ,J= 1 COOLOHE (J) ,J=1,NKPTRN) AND EXCHANGE POTENTIALS. UNITE (6,828) HRITE (6,824) NR1TE(6,820) READ (37) (VXCJNQ(J) ,DVXCDN(J) ,DPDN(J) , J= 1, NKPTil N)BEAD (27) (BUPO (J) ,DHUP (J) , DP'J P , (J) 1 = NKPTRN) J , READ KS2* IN AND A/2PI THE NCN-SEIF-CCNSISTENT IP (IETOTL.EO-0) GO TO 59 FOURIER eachan G t-cooif elation energy DSN=PBBAGA.DIOLA.DEN 129 130 DSN=PHEAGA.DIOLA.DEN UNITE(6,658) J=1,HAPTEN) UNITE(6,658) (DPUP(JJ, UNIT E(6,858) E(6,858) (VXCDNO(J),J«1,NKPTRN) UNIT IP (NSPIN. EQ. IJ GO TO IJ 884 TO GO EQ. (NSPIN. IP UNITE (6,860) UNITE (6,864) UNITE UNITE(6,870) UNITE (6, 856) (DVXCDH(J) ,J* 1,NKPTRN) 856) (6, (DVXCDH(J) UNITE UNITE (6 ,858) ,,858) J=1,NKPTRN) (6 (SDNO(J) UNITE UNITE (6,874) UNITE J*1,NKPTRN) , (J) (6,858) (DUDk UNITE (6,878) UNITE UNITE (fa,880) UNITE ,J»1,NKPTRN) (J) (6,E58) (DPDM UNITE UNIT E (6, £5 8) ,J=1,NKPTRN) 8) £5 (FDNO(J) (6, E UNIT UUPO [J) = HUPO(J)*2.DO/3.DO HUPO(J)*2.DO/3.DO = [J) UUPO DO 89 J* 1,NKPTNN 89 1,NKPTNN J* DO DHUP (J) = DBUP (J)*2.D0/3.D0 DBUP = (J) DHUP IF(NSPIN.EQ.2) GO TO 888 TO IF(NSPIN.EQ.2) GO CO ?? 2) EEXCH=IEXCH/4. EQ. (IETOT1. IF IF (IETOTL. HE. 2) GO TO 2) 89 TO GO HE. IF (IETOTL. ENERGY ELECTRON ELECTBCK SINGLE AVERAGE THE KBZ,EDf,NKBZPT,ESNGEL,CONST, CALCULATE CALL SUMOVK(EUP,IPT,IPTOL,KBX,KEY, 890 TO GO (NSPIN.EQ.1) (1)*DPDN(1) IF »DPUP 1) 0( PDN ♦ PO=POPO{1) ESNGEL=ESNG£L+ESNGED UNITE (0,894) UNITE NKBZPT,NKPTRK,ND,IETCT1,AC0NST,FENBI,£SNGEI, VP1 = 0.DO 0.DO = VP1 VICP1=0. DO DO VICP1=0. CALL SDNOVK(EDN,ZFT,IFTCL,KEZ,RET, KEZ,EDN,NKBZPT,ESNGED,CONST, KEZ,EDN,NKBZPT,ESNGED,CONST, CALL SDNOVK(EDN,ZFT,IFTCL,KEZ,RET, BY CONSIDERING ACRE KS POINTS. ANC CALCULATE THE TOTAL ENERGY. TOTAL THE POINTS. CALCULATE KS ANC ACRE CONSIDERING BY 2*0.DO VP VXCPD1=0.DO VXCPD1=0.DO CALCULATE D,U AND EXCHANGE-CCOPELAT1CN ENERGY AND THE CORRECTIONS CORRECTIONS THE AND ENERGY EXCHANGE-CCOPELAT1CN AND D,U CALCULATE UPUP1*0.DO UP 1*0. DO DO 1*0. UP VX CPU 1=0.DO 1=0.DO CPU VX GFE FKI , N3N0) , FKI GFE t'IETOTL*' ,11,2 X,'LATTICE. • ,F7.4,2X, CONST.* X,'LATTICE. ,11,2t'IETOTL*' GFERfll, NQNO) GFERfll, £P0,OBEG A £P0,OBEG 6*FERH1 = ',F9.6,2X,*SIHG1E. ELE. E=* , F 13. 7,/, 2X, 7,/, 2X, 13. ',F9.6,2X,*SIHG1E.F , = ELE. E=* 6*FERH1 fi'DENST OF K0=*,F9.7,2X,'CELL OF V*',F8.4,/)fi'DENST 89 89 CONTINUE 858 FORMAT (1 X.10F13.6) (1 858 FORMAT C(J), J=1,NKPTFN') /AT( , IX,'VXCDN 860 FORM 874 FORMATf/, 1X,'DHDN(J) , J=1,NKPTRN') 1X,'DHDN(J) 874 FORMATf/, 864 FOfcKAT(/, 1X,'DVXCDN(J) , 0=1,NKPTRN') 1X,'DVXCDN(J) 864 FOfcKAT(/, 870 FOEfJ AT(/, 1X,*UDN0(J) , J=KKPTBN*) t, 1X,*UDN0(J) AT(/, 870 FOEfJ 878 FOBM AT(/,1 X,* DEN SDN (J ) , J=1,NKPTRN') SDN DEN AT(/,1 X,* 878 FOBM 880 FONN AT ( / , IX,' DDENSDN (J), 0*1,NKPTRN*) ( / , IX,'AT DDENSDN 880 FONN 884 884 CONTINUE 888 888 CONTINUE B94 FOHMAT (///, IX , 'NO. K-* ,13,21, * NO, KS=•,13,2X,•NO, ,13,21, NB=',13,2X, 'NO. (///,, * K-* IX FOHMAT B94 890 890 CONTINUE U U U U U U U UUU U U (J u 131 DSN-PHBAGA.DICLA. DEN NPDN1=0.DO VCBSUB=O.DO VTSUH=0.DO VEPSDfl=O.DO VTPS0fl=0. DO VX CPS!1-0t DO VXCPU5=0.DO VXCPDS=O.DO opsun=o. DO NPUPSR=O.DO bPDNSfl=0. DO NPT= 0 696 CONTINUE IP (NSPIN.EJ.1) GO 10 9C0 BEAD (11,END=93 4) KSQ,GCCLT.GEBLD,GFXCH,GEXUP,GEXDN,GDSPA.GDSUP, £GDSDN,GDSUD,GD5DU,GDt1DP,GUBDN,KX1,KI1,KZ1,NNB IF (NPT.EQ.O) ERfilTE(6,SOU) KSQ,GC0L1,GEBLD,GEXCH.GEXOF,GEXDN,GDSPA,GDSOP, £GDSDN,GDSUD,uDSDU,GUBUE,GDREl.,KX 1,KY 1.KZ1, NNB IF (IETOTL.ME.1) GO TO 900 READ (12, END=934) KSQ.GCOLT, GE5LD.GN , GN'JP, GNDK, GDSPA, GDSUP, EGDSDN,GDSUD,GDSDU,GUBUP,GURDN,KXl,hI1.KZ1,NNB IF (NPT.EQ.O) CNF. ITE (6, 904) KSQ,GC0LT,GE1 DSN=PH3AGA.DI0LA.DEN GUn=GUR0P+GU3DK 91a CONTINUE 1P(NPT.NE. 1) GC TC 918 CCOULT=GCOLT GCCDELsGCOULT HRITE (6, 85B) GCOUE1,GCOULT,VKO(1) 916 CONTINUE IF (NPT. EQ.1) GO TO 920 GCOOLT=GCOtT GCOUEl>=GCOULT+ E.D0*PI* Z/OBEGA/AK52 920 CONTINUE IF (NPT.EQ.1) HE1TE (6,858) GCCUEL.GCOLLT VCESUfl=VCESUN*GCCUEI*ANNP VTSUfl=VTSUH*GCCOIT*ANNE VTPSUfl=VTPSU.*!*GCOOLT*GUB*AKNB ¥EPSUft=VEPSD1+GCOUEl.*GUe*ANND IF (NSPIN.EQ.2) GO TO 924 VX CPSfl=V XCPSB + GEXCH*GUfl*ANNB VXCPUD=*VXCP5B HPSUB=NPSDR+GW*GUfl*ANNB NPDDS* UPSDB 924 CONTINUE IF (NPT.LE. NKPTBN) GO TC 926 VP1=VP1*GCOUEL*CU«*ANNE VP2=VP2*GCOULT*GUB*ANNB IF (NSPIN. EQ. 2) GC TO 923 VXCP1=VXCP1*GEXCb*GUB*AK N B VX CP 2*VXCP1 HP1=NP1 + GH*Gl!N*ANNB HPUD2*NP1 928 CONTINUE IF (NSPIN.EQ. 1) GC TO 930 VXCPUS*VXCPUS*GEXUP*GUBUP*ANN3 VXCPDS* VXCPDS+GEXDK*GUEDK*ANNB VXCPUD=VXCPUS*VXCPDS NPUPSN=HFUP5N*GkUP*GUBUP*AKNB NPDNS«=HPDNSH*GHDN*G0BDN*ANN3 VPUDS*NPUPSN*H FDKS E IF (NPT.LE.NKPTBN) GO TO 930 VXCPU1=VXCPU1*GEXUP*GUKU F*AN KB VXCPD1*VXC?D1*GEXDN*GUBDN*ANN3 VXCP2* VXCP'J 1 * V XCFD1 NPUP1 = SPDPl*GBUP*GUItUP*ANNE UPON 1*K?DN1*GWDN*GUHDN*ANNB WPUD2*HPUP1*HPDN1 93 0 CONTINUE IF (NPT.EQ.1) HBITE (6,949) IF (NPT.EQ. NKPTBN.OB. NPT. EQ. 1.CR. NPT. EQ. 2.OB.NPT. EQ.300.OB. CNPT.£ Q .1000.OB. NPT.EQ.1500.OB.NPT.EQ.1800.CB.NPT.EQ.2000) CMITE (6,950) NPT, VF1, V E2 , V XCP2,KPUE2, VCESUB, VT SUB, VEFSUfi, VT PS’JB, 6VXCPUD,HFUDS GO TO 898 93 4 CONTINUE BEHIND 11 IF (IETCTL. HE. 1) GO TC 53B BEfcINC 12 93 8 CONTINUE VP*VP 1*0BEGA/2.DC IF(IFTOTL.BQ.2) NP*NP/4.D0 C 9 CONTINUE 99 nnnnnn non 5 FOEfl954 AT {1 X,2F12. 6 ,2 F11. 4, F9. 5, 16.F 7, 4F1 3. 6) 4 FORM944 AT 'ESBVEPIX, (//» 'ESBVCP 2X, 'NPT*,!!, , FR40' FH40',2X, 4 F0BBAT(//,2X,'NPT*,4X,'SEVEP940 X,'SBVCP FB40*,2 FE40*,2X, 4 FORM940 *AT(//,2X, NPT1,2X,' ESBVEP FR40* ,2X,•ESBVCP FR40',2X, 95 FORM 0 AT(IX,14,1X,6F12.7,4F11.7) 8 CHT/3,EE CI,X' SNL•7,D,1,D ,8X, SINGL.•,7X,•D•,11X,•D• FCRHAT(/,3X,'E1E 98 CCUI',3X,'E ' EC. X1 TTL,X' PTN.X KI»E1.',8X, E ' POTENT.4X, EXCH.' 4X,1E T0TAL',5X,'E G'E , G*ESBHP'UKUC-NUC') ,2X,'C*2*P1/A',3X, FF40' G M R I T E ( b ( 9 4 4G ) NR ITE (6, 94 0} Q • v3KFS<:riF i r r i * I G'SBVXP FR40*,2X,'SBWP 4X,«SBVEX',7X,•SBVCK• FR40• , 6X, . GNRIIE NFT,VP,VTP,HP,C,UNUNU,B (6, 950) FSUB, KPUP5K,KPDN S.1 G'SHHUP*PUP*,3X,'SBNEN*G'ESBNP X,•UNUC-NUC',4 FR40*,2X,*C*2*F1/A' 3 , EDM* SBBF*PP',4X, X, • ) G * S I I V E K * P K ' , ,3/ * 5 F . V C K * P K * ,3X , • S B V X K * P K * , 3 X , ' S B N K * P K •,/) GNRITE NIT,VF,VTF,HP,C,UNBNO (6,950) ESINGD (NE)x 0. DO IF (NDENS.EQ.O) GC TO 180 TN DN (KE) *0. DO SVUP (NE) =0.D0 IF (NSPIN.IF EQ. 2) ThUP(NE)= DO . S0 M D N ( N EEh ) =0. (NE)=EE D ODO NE*=1,itEHAX 100 IT (MSPI3. EQ.2) EE-EBIN IF (N5P1N. EJ. 1) EVEC5B=EVECSS/4.D0IF(IETOTL.EQ.2) E V E P S B = V E P S U B * C f i £ G D=-Z*VCESUB/2.D0+C*Z**2/2.D0/PI**2 A / 2.D O C=-11.432877 *2.DO*FI/ACONST CALCULATE THE DENSITY OF STATES IF (NSPIN. EQ.1) EPOTEH=D-U X = V X C P U U * O B E G A UNUNU=Z**2*C/(2.D0*P1**2) V X P = VTP=VP2*OBEGA/2.D0 V X C P 2 * 0 B E G A PB5SSH=(2. D0*EKIhET+EPCTEH*3.D0*lV|CSB)/E A 1 N E T = E S N G E L * 2 (3. . DEDO* 0 T * O U - T X H * EOB S N G E LEG + bC * A U - L EA) V C E U C L S B A TTH ETO E T AEK L E B G Y HP. ITE (6, 954) 2VEPSB,ESNGEL, U,D,EVEC£K,ETC7H,EPOTEH,EKINET, X, PRESSH HRITE (6,96) l i P = U P U D 2 * O M E G A E V E C S B = 8 P U D S * O M E G A D=-VTPSUB*O»2GA/2-E0 DSN-PHBAGA.DIOLA. DEN 133 no oo o 0 CONTINUE105 0 CONTINUE103 1 CONTINUE111 0 I E.TE) GO IF 104(E3.LT.E4) TO 105 0 CONTINUE100 0 I E.EE) GO IF TO102 104(E2.1E.E3) CO1 0 1 N T I N O S CNFITE ZERC,C41,11,E2, ,E4 E3 (6,140) FE.TE) GC TO2F(E1.GT.E2) 101 EE*E 1 GC TO 112IF (El.LE.E2) IF(D43.LE.ZERO) E4=E4*ZEFO E4*E0N (NB,I4) E2*E0N(NB,I2) E1*EDN (NB,11) CALL E4,COAST,EN,ESINGU,NEHAX) TELEC1(£l,E2,E3, CALL TD2NST(21,E2,E3,E4,EN,SHUP,NEKAI,CCN,ZERO) .DO*ZERO) .3 IF(E4.LE.E3.3B.E3.IE.E2.CR.E2.LE.Z1.0R.041.IE IF (041.LT.3.DO*Z£BO) E4*E4«ZEBO IF (021-EC.ZERO) IF(032.L£.ZERO) E2*E2-ZEEO £2*E3 EI*E2 IF (NSPIN. EO. GO1) TO 118 CALL TELECT(E1 E2,E3,E4,CON , ST, EN,TNUE, NEBAK) 041=E4-£1 I32*E3-22 £4 = EE E£=E3 F(1L.2 GCIF (E1.LE.E2) TO 102 13=IPT (3,IT) 12*1PT (2,IT> E3*ECN(KB,13) IF (021.LT.ZER0) D 2 1 = E 2 - E 1 IF (E2«GT.E3) GO TO 103 E3 = E4 E2 =2 E E£*E 1 F O B L L O K I NST G A T E B E NAR T S US E EFO DSU E B R O U T I NID E EDO l NB*1 l 118 NDFEEK , S T IP (EN (NE).GE.FEHHl.OB.EE.LE. GC FEE9I) TO 100 EE=E2 C43=E4-£3 E1=E2 E4-EUP (NB,I4)E3*EUP (NB,I3)E2=EUP (KB,12) E3*EE E1=EUP (NB,I1) IU*IPT(4,1T) I1=IPT(1,1T) IPTOLDO IT*1, 118 EE*EE*DE E S I N G D ( N E )= 0 . D C A N ( N E ) * F E R H I £1*E1-2.D0*ZE2O E1=I1-ZEFO D5 N=PBBAGA.CIOLI.DEN 134 o o n 1 3CO 0 N T I N U E 4 ULITE145 GEN.GSNDN.GTNDN (41 ,148) 6 FP T /»1,EjO,DNIYX LC O, X,*ESIN3L*,/) ILEC NO',6 ' FOPH160 AT{//»11X,'E'jIOX,'DENSITY7X, CONTINUE150 4 NKITE(4U,143 GEN,GS«UP,GTKUP 140) 2 FGF8AT EONS•120 UP',8X,'DEN EL 6X, • . (//,6X,'E',10X,*DEN EC UP*,7X,*£LEC CONTINUE IIS 4 FOBHAT148 (JF15.'j) 4 FOLHAT140 (1X,8F14.7) 1 I(3 TE) GO LT.E4) IF(E3. 114 TO 115 1 I(2L.3 GO TO 114 IF(E2.LE.E3) 112 1 CONTINUE115 1 CONTINUE113 CNP1TE EN (6, 140) (NE) ,SN0P (NE) ,StiDN{ NE) ,T«UP(NE) ,T«DN (NE) , EN0,E9AG fcHtiIT£(b, ZERO,E41,E1,E2,E3,E4140) fcDOWN*,5X,'TOTAL ELEC• S, ES1NGE GO TO 180 G S V DGEN=EN N E S N D N (NE) ( N E ) IF (DABS ESINGFs(S ESINGiJ(NE) +ESINGD (NE) IF DSN-PHBAGA.DIOLA.DEN IF (EE.GE.E4) GC TO 130 IF (EE.LT.E2) GO TO 100 IF (EE. LT.E3) GC TO 110 DE4-EF-E4 EAVBG- (El*E2*£3 + EE)/4.DO S -1.D0*CS4*DE4**3 GO TO 120 100 CONTINUE DE1-EE-E1 EAVBG- (E1*3.D0*EE) /4.D0 S=CS1*DE1**3 GO TO 120 110 CONTINUE DE2= E E-E2 DE22-DE2*DE2 DE23=DE22*DE? EAVBG- (E1+E2+2.D0*EE)/4. DO S=C5 23*DE23*CS 22-DE22 + CS21*DE2*C S2 0 120 S» (NE)-SH(NE)*S*COKST*EAVRG GO TO 140 130 EAVBG-(E1*E2*E3+E4)/4.DO SU (NE) = Sli{liE) + CCKST*EAVEG 140 CONTINUE RETURN IN D //LKED.S XSLIB DD DSN=D1103.CAILAUAY.RVCflPLID,D1SP-SHR / / DU DSN-D1103.CALLAWAY.BNDPKG.SU3LIB.CONPL,DISP-5HR / / DD DSN-SYS1.VfOBTLIB,DI5P=SUR / / DT DSN=S¥S2.FCBTLIE,D1SF=SIIB / / DD DSN=STS2.SSP.LIB,D1SP=SHR / / DD DSN-SYS2.PLOT.LIB,DISP-SHB //GO.F101F001 DD U.HT-3380 ,VCL=SER=USER77, DISP-SHR, / / DCB- (ELKSIZS=7294,RECFH-VSB,LREC1*00176), / / DSNAflE- PHB AGA.D10IA.rEFSP4.ECC.EAND //G0.FT1 1F001 DD UNIT-3360 , VCL-SEH-DSER77, E1SP-SBP, / / DSNARE-PHBAGA.DI0LA.FEF5P4.BCC.VK1 //GO.FT12F001 DD UNIT-3380,VCL=SIfi=USER77,EISP-SBP, / / DSKARE-PHEAGA.DI0LA.FEF5P4.BCC.VK3 //GO.FT4OF001 DD UNIT-3380,VCL-SEP-USER77.E1SP=(NEfc,CATLG), / / SPACE-(TRK, (20,10).ELSE), / / DCB- (ELKS1ZE*4500,BECFB=FP,IRECL=0045), / / DS.IANE- PltE AGA. E101A. FIF5P4. ECC.ENSTUP3 //G0.FT4 1F001 DD BSIT-338U,V0L-5EB=USBB77,DISP-(NEB.CATLG) , / / SPAC2-(TRK,(20,10),ELSE), / / DCD- (BLKSIZF=U500,RECFB=FE,IBECL=0065), / / DSNARE-PBBAGA.DIOLA.FEFSP4.BCC.DNSTDK3 //GO. FI17F001 DD UHIT-3380,VCL=SER=USER77, DISP-SIIR, / / DSNARh-PHBAGA.DlOLA.FEFSP4.BCC.BEIF71 //UO.FT27F001 DD UNIT-3380,VCL-5 FR-U5EB77,DISP-SHR, / / DSNA3I-PH2AGA.DIOLA.FEF5P4.BCC.SELFKK //GO.FI37F001 DD UNIT-3380,VCL-SES-USER77,D1SP-SHB, / / D5NAH E-PUBAGA.DIOLA.FEF5P4.DCC.SELF DD //GO.SIS1K DD * 1 1 901 20 1 2 2 15 0 43 1 40 1 •1.0 0.0025 0.00001 26.0 26.0 5.4057 // APPENDIX B 138 Table 81. Valence band energies at high symmetry points of theBrillouin zone of BCC iron - in Rydbergs (Ry) - for up ( +) and down (+) spins. The Lattice constant a = b.2 a.u. The Fermi energy Ep = -0.0398 Ry r+ T4- P+ P+ -0.6971 (1) -0.6811 (1) -0.3071 (4) -0.2186 (4) -0.2267 (25*) -0.1069 (25') -0.3071 (4) -0.2186 (4) -0.2267 (25') -0.1069 (25') -0.3071 (4) -0.2186 (4) -0.2267 (25') -0.1069 (25*) -0.0905 (3) 0.0727 (3) -0.1101 (12) 0.0484 (12) -0.0905 (3) 0.0727 (3) -0.1101 (12) 0.0484 (12) 0.6342 (4) 0.6999 (4) _- s Nt N+ H+ H+ -0.4429 (1) -0.3635 (1) -0.4354 (12) -0.3300 (12) -0.3185 (2) -0.2107 (2) -0.4354 (12) -0.3300 (12) -0.1078 (1) 0.0011 (l'J -0.0196 (25‘) 0.1198 (25') -0.0935 (4) 0.0419 (1) -0.0196 (25') 0.1198 (25') -0.0153 ( I 1) 0.0691 (4) -0.0196 (25’) 0.1198 (25’) 0.0037 (3) 0.1455 (3) 0.7285 (15) 0.7515 (15) co to Table B2. Valence band energies at high symmetry points of the Brillouin zone of BCC iron - in Rydbergs (Ry) - for up ( t) and down (+) spins. The Lattice constant a = 5.4057 a.u. The Fermi energy Ep = -0.1214 Ry rt r+ P+ P+ -0.7275 (1) -0.7125 (1) -0.3551 (4) -0.2689 (4) -0.2863 (25*) -0.1565 (25*) -0.3551 (4) -0.2589 (4) -0.2863 (25*) -0.1565 (25*) -0.3551 (4) -0.2589 (4) -0.2863 (25') -0.1565 (25*) -0.1729 (3) -0.0011 (3) -0.1880 (12) -0.0202 (12) -0.1729 (3) -0.0011 (3) -0.1880 (12) -0.0202 (12) 0.4931 (4) 0.5605 (4) N+ N+ Ht Hi -0.4725 (1) -0.3870 (1) -0.4556 (12) -0.3387 (12) -0.3618 (2) -0.2437 (2) -0.4556 (12) -0.3387 (12) -0.1869 (1) -0.0791 (1') -0.1121 (25') 0.0378 (25') -0.1752 (4) -0.0276 (1) -0.1121 (26') 0.0378 (25') -0.0936 (3) -0.0038 (4) -0.1121 (25') 0.0378 (25‘) -0.0927 ( I1) 0.0595 (3) 0.5863 (15) 0.6086 (15) Table B3. Valence band energies at high symnetry pointsof theBrillouin zone ofBCC iron - in Rydbergs (Ry) - for up ( t) and down ( + ) spins. The Lattice constant a = S.6 a.u. The Fermi energy Ep = -0.1817 Ry rf r+ Pt P+ -0.7433 (1) -0.7286 (1) -0.3881 (4) -0.2842 (4) -0.3291 (26') -0.1894 (25') -0.3881 (4) -0.2842 (4) -0.3291 (25') -0.1894 (25*} -0.3881 (4) -0.2842 (4) -0.3291 (25') -0.1894 (25') -0.2325 (3) -0.0532 (3) -0.2442 (12) -.0683 (12) -0.2325 (3) -0.0532 (3) -0.2442 (12) -0.0683 (12) 0.3849 (4) 0.4540 (4) Nt N+ Ht H+ -0.4908 (1) -0.3992 (1) -0.4647 (12) -0.3355 (12) -0.3917 (2) -0.2631 (2) -0.4647 (12) -0.3355 (12) -0.2442 (1) 0.1358 (1') -0.1802 (25') -0.0204 (25') -0.2342 (4) -0.0761 (1) -0.1802 (25') -0.0204 (25') -0.1638 (3) -0.0553 (4) -0.1802 (25') -0.0204 (25') -0.1492 (1’) -0.0016 (3) 0.4761 (15) 0.4983 115) Table B.4. Valence band energies at high syiaaetry points of the Brillouiri zone of FCC iron , in Rydberg (Ry),for up (t) and down (+) spins. The lattice constant is s = iS.5516 a.u. The Fermi energy is Ef * -0.0573 Ry rt r+ xt wt W 4 Lt L+ -0,6912 -0.69B2 (1) -0.4402 -0.3906 (1) -0.3630 -0.3095 (2 1) -0.4302 -0.3897 (1) -0.2143 -0.1357 (25M -0.3911 -0.3316 (31 -0.2756 -0.2162 (3) -0.2234 -0.1436 <3) -0.2143 -0.1357 125') -0.0379 +0.0614 (2) -0.2756 -0.2162 <3) -0.2234 -0.1436 <3) -0.2143 -0.1357 (25'J -0.0096 +0.0871 (5) -0.1183 -0.0302 (1) -0.0458 -0.0531 (21 -0.1111 -0.0178 <121 -0.0096 +0.0871 (5) -0.0095 +0.0873 (111 -0.0317 +0.0651 (3) -0.1111 -0.0178 <12) ■K). 1759 +0.1713 (4 1) +0,6171 +0.6439 (3) -0.0317 +0.0651 (3) -0.3762 -0.3208 ill -0.3418 -0.2866 (1) -0.1801 -U.1197 (3) -0.0960 -0.0044 (4) -0.0364 0.0580 (2) 0.4636 0.4952 (1) ■g» ro Table B.5 Valence band energies a t high symmetry points of the B rillouin zone of FCC iron , in Rydberg (Ry),for up (t) and down (+) spins. The lattice constant is a = 6.8107 a.u . The Fermi energy is Ep = -0.1283 Ry. rt rf Xt X4 W, Lt -0.7239 -0.7210 (1) -0.4645 -0.3921 (1) -0.4004 -0.3244 (2') -0.4639 -0.4059 11) -0.2738 -0.1718 (25*) -0.4213 -0.3395 (3) -0.3262 -0.2461 (3) -0.2823 -0.1776 (3) -0.2738 -0.1718 125') -0.1287 -0.0029 (2) -0.3262 -0.2461 (3) -0.2823 -0.1776 (3) -0.2738 -Q.171S (2511 -0.1022 +0.0188 (5) -0.1951 -0.0812 (1) -0.1227 -0.1205 (2 1) -0.1900 -0.0705 (12) -0.1022 +0.0188 (5) -0.1021 +0.0189 11’) -0.1217 +0.0001 (3) -0.1900 -0.0705 (12) +0.0767 +0.0816 (4’) +0.4736 +0.5135 (3) -0.1217 +0.0001 (3) Kt K 1 -0.4099 -0.3320 11) -0.3825 -0.3062 (1) -0.2458 -0.1652 (3) -0.1768 -0.0591 (4) -0.1243 -0.0057 (2) +0.3315 +0.3767 (1) 143 Table B. 6. Valence band energies a t high symmetry points o f the B rillouin zone fo r FCC iron , in Rydberg (Ry), fo r up ( t) and down (+) spins. The lattice constant is a » 7.0 a.u. The Fermi energy is Ep - -0.1501 Ry. rt r4 X+ X + Wt W* Lt L+ -0.7234 -0.4820 -0.3658 -0.4268 -0.3067 (2 1) -0.7390 (1) (1) -0.4877 -0.3904 (I) -0.3176 -0.1537 (25 1) -0.4453 -0.3090 (3) -0.3626 -0.2318 (3) -0. 3246 -0.1607 (3) -0.3176 -0.1537 (25’) -0.1909 -0.0010 (2) -0.3626 -0.2318 (3) -0.3246 -0.1607 (3) -0.2483 -0.3176 -0.1537 (25 1) -0.1673 +0.0229 (5) -0.0745 (1) -0.1846 (3) -0.1526 (2') -0.2447 -0.0633 (12) -0.1673 +0.0229 (5) -0.1673 +0.0229 (l'l -0.1846 (3) +0.0041 (3) -0.2447 -0.0633 (12) -0.0183 +0.0368 (4 1) +0.3838 +0.4539 (3) -0.1662 (2*) +0.0041 (3) -0.4343 -0.3118 (1) -0.4123 -0.2850 (1) -0.2915 -0.1615 (3) -0.2326 -0.0537 (4) -0.1865 +0.0005 (2) -0.2492 +0.3278 (1) 144 145 Table B.7. Least square fit coefficients, as defined in section D of Chapter 4, for the charge form factors of BCC iron. The lattice constants used are 5.0, 5.4057, and 6.0 in atomic units (a.u.). c k Z m ac ac a0,k,Jfc,m 1 ,k,A,m 2,k,Z ,m 1 1 0 3.58136484 3.82979289 -0.204981172 2 0 0 -0.53493592 3.91177317 -0.188461197 2 1 1 -2.02307672 3.65074480 -0.158685891 2 2 0 -1.40850876 2.94225321 -0.099150292 3 1 0 0.20577533 2.05663904 -0.029130822 2 2 2 0.86298061 1.65095044 -0.003977313 3 2 1 2.01638714 1.12533471 0.031469571 4 0 0 3.33933146 0.57493513 0.069487715 3 3 0 3.02149173 0.65213636 0.053233058 4 1 1 3.43664824 0.49926231 0.066621608 4 4 2 0.78131382 1.23413160 -0.039602873 6 0 0 1.40023291 1.01385793 -0.020968903 Table B. 8 . Least square f it coefficients, as defined in section D of Chapter 4, for the spin form factors of BCC iron. The lattice constants used are 5.0, 5.4057, and 6.0 in atomic units (a.u.). s k £ m as as a0 ,k,£,m 1 ,k,£,m 2 ,k,£,m 1 1 0 -0.92334553 0.47537670 -0.034361518 2 0 0 -0.61031707 0.28057959 -0.016707236 2 1 1 -0.83298532 0.28955795 -0.016196177 2 2 0 -0.60380824 0.18832302 -0.008220275 3 1 0 -0.01999695 -0.00250778 0.00597343b 2 2 2 -0.61723276 0.16405534 -0.006977759 3 2 1 -0.26844538 0.05403997 0.000941820 4 0 0 0.46889063 -0.16349990 0.016636354 3 3 0 -0.13055622 0.00777395 0.003611459 4 1 1 0.23540831 -0.09688971 0.011253610 4 4 2 0.01297626 -0.02959130 0.004099209 6 0 0 0.41962568 -0.13630275 0.011227522 APPENDIX C 147 148 C.l. Evaluation of the Pressure for a Metal The expression of the pressure P is, for a paramagnetic OR substance. 3PV= 2Eki + EPC - 3 I " Ek1 “ S I 'rn(l‘>?) (' ,2) d3r’ C-1’2 Epr = -211 I d3r + / d ?d ?' + jj ■ I * 11- C .l.3 Pc " |M | I?-?'I ^ |S4v| V is the crystal volume. The remaining terms in equation C .l.1 are as defined in Chapter 2. Using equations 1 and 2 and the definitions of U and D in Chapter 2 one readily obtains: T = Z En(ic) + 2U - Z I p Vxc d r C.1.4 The above expression of Ep^ yields: Epc = D - U C .l.5 The last term in equation C.1.1 (sign included) in simply &x(.. We therefore have: 3PV = 2 I E (it) + 3U + D - 2X + Axc, C .l.6 149 with This expression can be precisely evaluated by the new version of BNDPKG as shown in program Density provided in appendix A.3. As stated above, QC this new method does not use a Muffin-Tin or other approximations in obtaining the pressure P. VITA Diola Bagayoko, a son of Djigui Bagayoko and Nagnouma Keita, was born on December 12, 1948 in Bamako, Mali. He attended the secondary school of N'Tomikorobougou from 1962 to 1966 and the Prosper Kamara High School from 1966 to 1969, where he majored in mathematics, physics, and chemistry. He majored from 1969 to 1973, in physics and chemistry at the Ecole Normale Superieure de Bamako. Bagayoko earned a national reputation as a good student by systematically holding the l s^ rank in his classes from primary school to college. After two years of teaching physics and chemistry at Askia Mohamed and Sikasso High Schools he enrolled at the American Language Institut at Georgetown University to acquire english proficiency in October of 1975. He earned a Master's degree in physics from Lehigh University in 1978 and joined the group of professor Callaway at Louisiana State University the same year. A few of the awards and honors Diola Bagayoko accepted include but are not limited to: the Malien Government fellowship from 1966 to 1973, a French government fellowship (Bourse FAC) from 1970 to 1971, Operation Cross-road Africa Award to visit the United States of America in the summer of 1972, a University of Grenoble (France) grant for practical training in the summer of 1973, African graduate fellowship (AFGRAD) offered by the African American Institute, Brandeis and Lehigh Universities from 1975 to 1980, a Physics department teaching assistantship at Louisiana State University from 1978 to 1981 followed by a research assistantship from 1982 to present. 150 151 Diola Bagayoko has been holding membership and leadership positions in various student and professional organizations. These include, at the professional level, the American Physical Society (APS), the International Association for the advancement of appropriate technology for developing countries, and, at the social level, the Organization of Lycee Proper Students (President 1968-1969), the Organization of EN Sup Students (public relations officer 1970-1971), the African Student Organization of Louisiana State University (vice-president 1979-1980, president 1981-1982, and 1983 to present), the International Student Association at L.S.U. (vice-president 1980-1981, president 1981-1982). Bagayoko has been participating in several international physics conferences. Few of those at which he presented a publication are: the APS Annual meeting in New York City (1980), APS meeting in New Orleans, Louisiana in November 1981, the APS annual meeting in Dallas, Texas in March 1982, the Sanibel International symposium in Palm Coast, Florida, in March 1983. Bagayoko delivered invited talks at social conferences on such subjects as world hunger and voluntarism. He coauthored and/or authored numerous publications in a number of journals (i.e ., physics letters, Physical Review and the International Journal of quantum chemistry) on the electronic, magnetic, bulk, optical and other properties of metals as well as computational procedures. Detailed references are available in his Ph.D. dissertation. Married and father of one, Diola Bagayoko is currently a Ph.D. candidate in physics at Louisiana State University. His present research interests include the electronic energy band theory of metals and related topics, Laser physics, and the search of solution to relativistic quantum mechanical equations. EXAMINATION AND THESIS REPORT C an d id ate: DIOLA BAGAYOKO Major Field: Physics Title of Thesis: Electronic Structure of Iron Approved: [ajor Professor and Chairm. EXAMINING COMMITTEE: Date of Examination: July 15. 1983