Potentials of Interatomic Interaction in Molecular Dynamics

Rev.Adv.Mater.Sci.Potentials of interatomic 20(2009) interaction 1-13 in molecular dynamics 1

POTENTIALS OF INTERATOMIC INTERACTION IN MOLECULAR DYNAMICS

Alexander I. Melker

Department of Metal Physics and Computer Technologies, St. Petersburg State Polytechnic University, Polytekhnicheskaya 29, 195251, St. Petersburg, Russia

Received: February 05, 2008

Abstract. In this contribution we have analyzed the principles lying at the basis of interatomic potentials which are used in molecular dynamics studies. The different types of potentials were considered: ab initio, semi-empirical, and empirical. The classical, ab initio, and bond-charge molecular dynamics were discussed.

1. INTRODUCTION and then integrates numerically the equations of motion of N particles. At N >>1, the system has a Over the course of the past half-century Molecular statistical property. This allows to watch the mo- Dynamics has progressed from a small branch of tion of individual particles and observe dynamical scientific investigation to an independent science. structure of the system; and simultaneously calcu- The onset was done in 1955 when specialists in late the average properties of the system with re- computational mathematics asked Enrico Fermi to spect to time and ensemble of particles (energy, set them a task which must satisfy three condi- pressure, temperature, stress-strain diagrams, tions: etc.). The calculated averages can and must be • it must be interesting for physics, compared with experimentally observed values. • it has no analytical solution, Therefore, once the problem for study was put, • it is not beyond the capacity of existent comput- you have to choose the most appropriate method ers. of numerical integration of differential equations of A computer allows to obtain an exact numerical motion and the most appropriate law of interaction solution, which helps to understand the kernel of for the particles of a system studied. The methods the problem. After such a computer prompt, one of integration were considered thoroughly in [1]. can separate main and secondary factors, formu- Now we only mention that a real professional does late the problem in a new fashion, and construct a not rely on numerous commercial programs be- simpler nonlinear equation having an analytical cause ‘a person must know how the calculations solution. As a rule, at that, there appears a new are going on. If one does not understand how it is scientific direction with a lot of interesting compu- done, one sees only bare numbers, but their genu- tational and analytical results. Such new method ine significance is hidden in calculations’ [2]. Just of scientific investigation, the combination of analy- the same approach is valid to a greater extent for sis and computer simulations, becomes more and the choice of interaction law. more important in a scientific discovery. In this contribution we consider interatomic po- Molecular dynamics is a type of computer ex- tentials which were deduced from the first principles periments where one assigns the law of interac- as well as semi-empirical potentials and briefly tion for N particles, initial and boundary conditions, empirical ones. Corresponding author: Alexander I. Melker, e-mail: [email protected]

2. THOMAS–FERMI ATOM The one-particle Schrödinger equation has the solution The atom model suggested by Thomas and Fermi (L.H. Thomas, 1926; E.R. Fermi, 1927) has be- ψarkrf = expai f , come a classic and can be found in many books on quantum mechanics, e.g. in [3]. The model is at that based on two ideas: one of them is taken from h22k quantum mechanics and another from electrostat- Eafk = . ics. The starting point of the quantum mechanical 2m theory of an atom is Schrödinger’s equation for Here k is any vector which does not depend on electrons coordinates. Let us restrict the motion of electrons HEΨΨ= . by volume V. In this case Here Ψ, E is the wave function and the total energy ψaf2 = of all electrons respectively and the Hamiltonian z rrd 1 has the form V and hence p 2 1 e 2 =−j 2 + H ∑∑∑Ze ∑ . 1 j 2m j r i < j r ψaf= a f j ij rkrexpi . V The first term gives the kinetic energy of electrons, the second one interactions of the electron with a If we select the volume V as a cube with the side L nucleus, and the third describes interaction of the and put the periodic boundary conditions electrons with each other. In order to obtain the ψψaxLyz+=+=,,f a xyLz , ,f simplest description of electron motion, let us exclude the interaction of electrons with each other ψψaxyz,,+= Lf a xyz ,,f , and with the nucleus. Then we obtain the model of free electrons in which Schrödinger’s equation we exclude the influence of a surface. takes the form The periodic boundary conditions mean that an electron does not reflect going to a surface, but p 2 h2 goes out the volume and simultaneously enters the j Ψ∆ΨΨ=− = ∑∑j E . volume through an opposite plane, e.g. in one-di- 22mm j j mensional case the electron moves in a ring. By Let us write down the many-particle wave func- virtue of the boundary conditions tion Ψ as the product of one-particle functions ψ expaiikra += Lff expa krf , each of them being dependent on only one-electron coordinates i.e. .

Ψaf= ψ af a f = rr1,...,Nj∏ r . expikr 1 . j This means that Substitute this formula in the Schrödinger equation. The latter disintegrates into one-electron equa- kL⋅=2πn, tions where

h 2 n = 012, , ,... −=∆ψaf ψ af jjrrE , 2m or for which the sum of one-particle energies is equal F 2πI ===α to the total energy kααn, xyzn , , , α012 , , ,... H L K ∑ EE= . j are integers. j Consider how to interpret these results. The We will investigate only a one-electron equation, vector equation r . k = const defines a plane nor- so delete the index j. mal to the vector k. On this plane the function Potentials of interatomic interaction in molecular dynamics 3 exp(ikr) has a constant value, whereas along any can add any constant value, it is reasonable to ac- ε straight line, which is parallel to the vector k, the cept that max = 0. In this case function changes periodically. For this reason, the function exp(ikr) is termed a plane wave and the 2 2m k =− Uraf, vector k is referred to as a wave vector. max h 2 It follows from the boundary conditions that the wave vector can take only the values which and the problem is reduced to the question how to comppponents are integers of 2π/L. One can de- find U(r). fine these mathematically allowed values as quan- With this purpose, refer to electrostatics. Let tum numbers which characterize electron states in the electron density at the distance r from the the system. Let us introduce k–space and construct atomic nucleus is equal to n(r). Then the electro- ϕ the lattice in this space, the knots of which are the static potential (r) produced by joint action of the allowed values of the wave vector. Since the elec- nucleus and electrons satisfies Poisson’s equation tron energy is proportional to the wave vector (Siméon Denis Poisson, 1812) squared, the lattice gives all the possible values of ∆ϕ = 4πenar f, electron energy. Therefore, we can find the physical state of the system with the help of the geo- at this U(r) = -eϕ(r). metrical scheme developed. Now we have two equations which incorporate Suppose that the system contains N electrons. the electron density; one equation is quantum one The electron energy has a minimum in the ground and another is Poisson’s equation. Using the quan- state. To obtain this state, it is necessary to fill N/2 tum equation, one can exclude the electron den- cells of least energies, each cell having the vol- sity from Poisson’s equation and obtain so-called ume (2π/L)3 and containing two electrons with op- Thomas–Fermi’s equation posite spins. Since the energy does not depend on 32/ the direction of a wave vector, the filled cells cre- 42emea f ∆ϕ = ϕ 32/ . ate a sphere. The sphere is spoken of as Fermi’s 3πh 3 sphere and its radius k is known as the Fermi wave F It is reasonable to use the atomic units, i.e. to put vector (Enrico Fermi, 1926). Therefore e = m = h = 1. Then N af2π 3 N 4 33 82 ==4ππk , 32/ F ∆ϕ = ϕ . 2 V V 3 3π and hence the electron density of the system is The solution of this equation must satisfy the boundary condition in the immediate proximity to the 3 N k nucleus, where the potential ϕ(r) at r → 0 must n ==F . V 3π 2 coincide with the nucleus potential Ze/r, i.e. in the atomic units rϕ(r) → Z. Now introduce the potential energy U(r). Tak- Let us find the derivatives of the expression rϕ(r) ing into consideration this energy, one can write for any electron the following equality dafrϕ dϕ =+ϕ r , dr dr h 22k max +=Uraf ε . d2 afrϕ d2 ϕϕd 2m max =+r 2 . dr 2 drr2 d From this, it follows Since ϕ depends only on r, the Laplace operator 2m reduces to k 2 =−afε Uraf. max2 max h d2 2 d ∆= + , ε drrr2 d Here max is the maximal value of the total energy. The total energy is constant if the system consid- so ered is in a stationary state, i.e. the electrons do not change their positions in an atom. Since any 1 d2 arϕf ∆ϕ = . potential energy is defined as a value to which one 2 dr 2 4 A.I. Melker

Denote Z1e, e2 = Z2e and choosing k = 1, write down the interaction energy of two point charges Z e, Z e as rrϕχaf= Zr af 1 2 ZZe2 and rewrite Thomas-Fermi’s equation in the form UrZaf,, Z = 12 . 12 r πχ2 af 3 rZd 32/ = afZχ . Atoms are not point charges. Besides their nuclei, 2 82 dr which dimensions can be ignored, they have elec- Introduce the dimensionless variable x = r/a. Since trons. If the distance between the nuclei of both 2 2 2 atoms during an elastic collision becomes less than dr = a dx , we have atom sizes, the electrons begin to influence on both 23/ 13/ aZ===di3ππ// 8 2bg 92 128 Z nuclei in a similar manner. On the contrary, at a large distance between the nuclei exceeding atom − 0., 885Z 13/ sizes significantly, the electrons screen the nuclei and, as a result, the atoms do not interact. Conse- and we come to the equation quently d2 χ x = χ 23/ ZZe2 d x 2 UrZaf,, Z = 12 χ afrZ ,, Z , 12 r 12 χ χ with the boundary conditions = 1 at x = 0 and = 0 at x → ∞. where the screening function must satisfy the fol- The equation obtained has no parameters and lowing condition defines the universal screening function, but, at the R10, r → same time, the equation has no analytical solution, χafrZ,, Z → S . so one tabulates the function χ(x). The good ap- 12 T0, r →∞ proximation for χ(x) is

If Z1 = Z2 = Z one can take Thomas–Fermi’s 1 function as a screening one χafx ≅ , a1+ αxf2 χαa f ≅+a f−2 = = 13/ xxxraaaZ1,/,./, 0 885 0 where α = 0.536 (T. Tietz, 1956). αχ=≅+a a α−2 Return from the atomic units e = m = h = 1 to 0.. 536xxf 1 f . ordinary ones. Since Expanding the function in a Taylor series, one ob- == −13/ tains rax0., 885 Zax0 where a = h 2/me2 is the Bohr radius, we have 1 0 =−12ααxx + 3af2 −... af+ α 2 13/ 1 x Ze F ZrI ϕχafr = G J. H K At small x the expansion coincides very closely with r 0. 885 a0 the exponent expansion

3. ELASTIC COLLISIONS OF ATOMS expaf−=−+2122ααααxxx af2 − ..., 21071 =≅ . ,

Consider an elastic collision of two atoms. To char- so acterize the collision completely, it is necessary to know their potential of interaction. It is well known F r I F r I a χ ≅−exp ,a =0 . that the interaction of two point electric charges e , 13/ 1 H aK H aK Z e2 can be expressed by Coulomb’s law (Charles Coulomb, 1785): For different atoms Bohr (Niels Bohr, 1948) has suggested to take ee fk= 12. 2 13///=+23 23 r ZZZ1 1 . Here f is the interaction force, k is a constant de- The expression is proportional to the quadratic 1/3 pending on a unit system chosen. Denoting e1 = mean of the functions Z of both charges. It should Potentials of interatomic interaction in molecular dynamics 5 be emphasized that the exact numerical solution oms, bearing in mind Coulomb’s law, one assumes for different atoms was obtained by Firsov that (O. Firsov, 1957), and so Thomas-Fermi’s poten- aa+ tial with Firsov’s screening function is referred to ==12 AAAa12,. as Thomas-Fermi-Firsov’s potential. 2 Let r → 0. Then the electrostatic potential gen- At last, at very low energies of colliding atoms, the erated by joint action of the nucleus and electrons hard-sphere model is used, where has the form R∞<, rb bb+ Uraf= S b = 12. Ze F r I Ze Z 43/ T02, rb≥ ϕafr ≅−=−H1 K . r a r a0 We have considered the main basic potentials of interatomic interactions which one uses to ana- Here the first term is the potential of a nucleus, lyze elastic collisions. Besides, there are plenty of and the second term represents the potential cre- formulas which one uses to approximate experi- ated by electrons in the origin of coordinates. One mental data; these formulas are also named ‘po- can see that the electrons screen the nucleus, so tentials’. For example, if to take the Thomas-Fermi the interaction of nuclei screened is reduced in screening function in the form of exponent sum comparison with of bare-core interaction. χa f =−+−a f a f At large distance from a viewer (r >>a), a nucleus xxx035. exp 03 . 055 . exp 12 . together with electrons is a neutral atom. In many +−010. expa 60 .xf , cases such system can be considered as a whole, e.g. when the thermal vibrations of atoms in a solid one obtains Molière’s potential (G. Molière, 1947). The advantage of this potential lies in the fact that are studied. Besides, at r >>a the interaction of atoms is determined mainly by the screening func- the potential allows obtain results in an analytical tion. Let us leave in the screened potential only the form. “There are 66 ways to compose the songs of exponent. Then we obtain Born–Mayer’s potential tribes and all of them are right” (Joseph Rudyard (M. Born, J.E. Mayer, 1932) Kipling, 1865–1936, Nobel Prize for literature in 1907). Uraf=− Aexp a r / a f . For the first time, the exponential screening appeared in the theory of electrolytes (P. Debye, The authors used this potential to calculate prop- E. Hückel, 1923), the function U(r) = Aexp(-r/a)/r in erties of ionic crystals similar to NaCl. The coeffi- solid state physics has the name screened cients A, a were found experimentally on the basis Coulomb’s potential, in the meson theory it is la- of compressibility tests. It turned out that for all 17 beled as Yukawa’s potential (1935), at last, con- crystals studied the constant a was practically con- sidering transmission of charged particles through stant (variations were less than 6%), but the coef- the matter, it is spoken as Bohr’s potential (1948). ficient A changed in a rather large range. It is worth Independently of the name, in all the cases a sepa- noting that Born–Mayer’s potential does not follow rated particle is surrounded by the particle cloud of directly from a theory; it is a volitional decision, but opposite sign. The charge density decreases ex- this approximation allows describe experimental ponentially, so the constant a is named screening data rather well. length; in plasma physics it is referred to as Debye From the conditions length. EUrA==afexp a − raE / f , =µν2 /2 , 4. SEMI-EMPIRICAL POTENTIALS where µ is the reduced mass and ν is the relative velocity of colliding particles, one can find the mini- 4.1. Hydrogen atom mum distance between identical atoms during col- Consider an atom of the simplest element, hydro- lision gen, within the framework of classical mechanics. Hydrogen consists of a proton having the mass m ==a f =a f = p rbaAEaAEbrmin ln / ln2 /1 ,min . and an electron with the mass me. Both particles attract to each other according Coulomb’s law Here E1, E2 is the kinetic energy of relative motion and the kinetic energy of a projectile atom respec- e 2 tively. It is supposed that another atom (target atom) Uafr =− . was at rest before the collision. For different at- r 6 A.I. Melker

Here e is the electron charge, r = |r - r | is the 1 2 222222=+=+ϕ distance between the particles. The problem of mo- ddlxyrr d d d, tion of two particles both in classical mechanics mlF d I 2 m ==+bg& 222ϕ& and in quantum one can be reduced to the prob- Ekin rr . 22H dt K lem of motion of one particle with a reduced mass in the central field U(r) which is symmetric regard- From the conservation law of angular momentum ing an immobile center of coordinates. In our case we find me<

M = m xyz, Here xy&&&z M e 2 M Uraf=+ Ur af =−+ where i, j, k are the dimensionless unit vectors di- M 22mr 2 r mr 2 rected along the axes x, y, z respectively, we have is the effective potential energy. It follows that one

2 can consider the radial part of motion as a one- M≡= M maf xy&& − xy = mrϕϕ&& == J const. z dimensional motion in the field with an effective ≡=2ϕ& MMz mr. potential energy. A one-dimensional problem is simpler, but there appears the pseudopotential Here J = mr 2 is the moment of inertia. Thus to M/2mr2.This result is the consequence of the theo- describe two-dimensional motion in any plane rem (William Thomson (Kelvin), Peter Godfrey Tait, z = const, it is enough to use two polar coordinates 1879), according which any potential energy can ϕ r, . be considered as a kinetic energy of hidden mo- Write down the velocity square in the form tions that are inaccessible to an observer. ν2 ==afd/dlt2 d l22 /d. t 4.2. Pair potentials From this it follows that kinetic energy is propor- By analogy with a hydrogen atom consider a di- tional to an arc element squared in an appropriate atomic molecule in the framework of classical me- system of coordinates. In our case chanics. At a short distance the atoms repel that excludes their overlapping, but at a long distance Potentials of interatomic interaction in molecular dynamics 7 they attract to each other creating a molecule from Here σ = (A/B)1/6, ε = B2/4A, A, B are constants of separate atoms. Thus forming a molecule re- the Mie potential. The parameter ε gives the depth sembles two-body problem. Probably due to this of a potential well, the parameter σ defines the ra- analogy, Gustav Mie (1903) suggested to describe dius of repulsion and can be found from the condi- interaction in a diatomic molecule with the help of tion U(σ) = 0. potential similar to UM(r) When discussing collisions, we analyzed quite a few of repulsive potentials. Those potentials are A B used in molecular dynamics too. For example, com- URaf=−. R nmR bining the Born-Mayer potential with van-der-Waals attraction one obtains the Buckingham potential Here A, B > 0, n > m > 0. This function has a mini- (1947) mum at

nm− B RnAmB= /. af=−− a f 0 UR Aexp ar . r 6 In the early twentieth century the Mie potential was widely used for calculation of physical proper- We have considered only the most popular semi- ties of gases and solids. Then it was displaced by empirical pair potentials; however the number of the Morse potential suggested potentials is very great. Strictly speak- ing, the efforts to find the best potential on the ba- af=−−k α a f UR U00exp 2 R R sis of experimental macroscopic data are inefficient −−−αafp because this inverse problem refers to the class of 22expRR0 . incorrect ones. In mathematics incorrectness

This function has the minimum -U0 at the point R = means that either a solution does not exists, either → ∞ R0 and tends to zero when R . Suggesting this it is not unique, or the problem is unstable (Jacques potential Philip M. Morse (1929) proceeded from Salomon Hadamard, 1923). the following condition. The potential being substituted into Schrödinger’s equation must describe 4.3. Three–body interaction exactly allowed energy levels of a diatomic molecule. The only region where the Morse potential Consider the problem of structure and stability of does not agree with numerical calculations is in microclusters. Here the crucial question relates to the vicinity of R = 0. Here the numerical solution their nucleation and growth. The corresponding tends to infinity whereas for all 21 molecules stud- calculations are divided into two groups: atomistic 2 4 methods which, in general, employ two-body po- ied by Morse U(0) falls in the range 10 - 10 U0. However, these values are so large that they do tentials and typically predict compact three-dimen- not influence on wave functions and energy levels. sional configurations as the energetically more More details can be found in [4]. stable ones; ab initio calculations for small metal The simplest solids are crystals of inert gases clusters which often predict that two- or even one- with closed electron shells. Even when their atoms dimensional configurations may be energetically approach the minimal distance at which crystalli- preferred. Since the ab initio calculations are more zation takes place, the electron wave functions of reliable, the discrepancy means that pair poten- neighboring atoms do not overlap and the atom tials are inadequate for investigating this problem. attractions remains of van-der-Waals nature. In the The exit was found when both two-body and three- thirtieth years of the twentieth century Lennard- body interactions were incorporated into the po- Jones did numerous calculations of kinetic prop- tential energy of microclusters (T. Halicioglu, P.J. erties of inert gases using the Mie potential with White, 1981). m=6. It turned out that experimental data could be The energy of interaction for a system of N par- equally well described, if one chose n in the inter- ticles can be expressed as val from 9 to 12. For convenience of calculations, 1 1 it was accepted that n=12. The 6-12 potential is =+∑∑∑ bg∑∑ b g Uurrij,,,, u rrrijk called the Lennard-Jones potential and is written 2 ii,,≠≠ j j 6 ii≠ j jk k in the canonical form as where u(ri,rj) and u(ri,rj,rk) represent two-body and LF σσI 12F I 6 O three-body potentials, respectively. A Lennard- URaf=−4εM P. Jones type two-body potential was assumed to NH RRK H K Q describe u(ri,rj). The triple-dipole Axilrod-Teller po- 8 A.I. Melker tential (B.M. Axilrod, E. Teller 1943) was consid- the electrons, the nuclei are moving so slowly that ered for approximating the three-body interactions the electrons feel only the nucleus positions, but u(ri,rj,rk): not their velocities. Since on the motion of electrons only the field of immobile nuclei has influ- 13+ cosθθθ cos cos ucbgrrr,, = ijk. ence, one can write the separate Schrödinger equa- i jk 3 b g3 tion for the electrons rrrij ik jk ψψaf= afaf Here c3 denotes the triple-dipole interaction con- HrRERrRel;;, el stant which depends on the atomic species involved ψ in the interaction and reflects the intensity of the where (r;R) is the eigenfunction of the electronic θ θ θ Hamiltonian three-body interactions; i, j, k and rij, rik, rjk are the angles and sides of the triangle formed by the p 2 three particles I, j, and k. =+∑ e af,. Hel UrR It turned out that the microclusters were three- e 2me dimensional for small values of c . With an increase 3 The eigenvalue E (R) is the value which would be in the intensity of the three-body interactions there el equal to the total energy if the nuclei having the was a transition to, at first, two-dimensional and geometrical arrangement R created an electrostatic then to one-dimensional forms. field and mutually repelled, but did not move. In other words, the electron wave function ψ(r;R) de- 5. EMPIRICAL POTENTIALS pends on electron coordinates, but nuclei coordi- 5.1. Born-Oppenheimer’s nates are parameters, so that the electron energy

approximation Eel(R) becomes a function of these parameters. In order to separate the variables r and R, one The adiabatic approximation was developed by can apply the Fourier method which was sug- Born and Oppenheimer (Max Born, Robert gested, developed, and formulated by the follow- Oppenheimer, 1927), but is rarely referred to by ing scientists: Jean Le Rond D’Alembert (1749), their names. It can be found only in some books Jean Baptiste Joseph Fourier (1821), Mikhail on quantum mechanics, e.g. in [3]. Since a mol- Vasilievich Ostrogradskii (1826). Suppose that the ecule consists of electrons and nuclei of atoms, solution of the full Schrödinger equation can be the Hamiltonian of a molecule includes the kinetic written as a product of two functions energy of electrons and nuclei, the interaction energy between them and the energy of interactions ΨarR,;.f =ψa rRf χa Rf with external fields. Having excluded the interaction with external fields, write down the Hamilto- Then nian of a molecule in the form af+= afψχ a faf ψχ a faf Hel r H n R rR;;, R E rR R

22 p p where H =++∑∑e n UrRaf,.

e 22me n M n P 2 HRaf= ∑ n . Here pe, Pn is the linear momentum of an electron n n 2M n and a nucleus, respectively, me, Mn is, as before, the mass of an electron and a nucleus, r is the set The operator Hel(r) differentiates only with respect of coordinates of all electrons, R is the set of coor- to electron coordinates, therefore dinates of all nuclei, and U(r,R) is the total potential energy, i.e. the sum of electrostatic interactions a fψχa f a f = χa f a f ψa f Hrel rRR;; RHr el rR between all the particles. This energy depends on = χψaREf a Rf a rR;.f the geometrical arrangement of electrons and nu- el clei which is unknown a priori. The corresponding As a consequence the full Schrödinger equation Schrödinger equation for such a system is takes the form HrRErRΨΨaf,,.= af a f +=a f ψχa f a f ψχa f a f Hnel R E R rR;;. R E rR R Suppose that the electrons are insensitive to The operator Hn(r) differentiates with respect to the linear moments of the nuclei. In other words, nucleus coordinates and, hence, acts on the both since the nuclei are at least ~104 times heavier than functions, electronic and nuclear. At that Potentials of interatomic interaction in molecular dynamics 9

22 '' '' ' ' '' netic energy which possesses additivity. If the sys- PR~/∂ ∂ ,af ψχ=+ ψ χ2 ψ χ + ψχ . tem is in an external field, each ion is under the If the electron wave function changes very slowly action of the force which depends on the external at the change of nuclear coordinates, then its de- force potential. This potential is also additive, so rivatives with respect to the parameter R are small. = ext Therefore HUext∑ i . i aψχ f'' ≅= ψχ'' afa ψ faf χ The second term gives direct interaction of the ions; ,;HRn rR R the third defines indirect interaction through valence ψχarRH;.f a Rf a Rf n electrons. Both terms can be represented in the Having substituted in the full equation, one obtains form of the many-body potential

a f +=a f χχa f a f U aRR, ,..., Rf , NN12 HRnel ER R ER. Thus instead of one Schrödinger equation for elec- which depends on an ion configuration, the con- trons and nuclei, we have now two equations: one figuration being unknown a priori. for electrons and another for nuclei How to solve the problem? Suppose that the many-body potential depends on the coordinates ψψaf= afaf U of two particles only, and this dependence has the HrRERrRel;; el V. form HRaf+= ER afafχχ R ER afW nel afaf==∑∑∑ − ∑ . UUUNNRR12, ,..., R R ii R ij Both equations are known as Born–Oppenheimer’s i <

σε= c . One can associate the elastic constants not only ij ijkl kl with a type of deformation but also with an inter-

Here the coefficients cijkl form the tensor of elastic atomic potential. Suppose that all the atoms of a constants. As this takes place, the density of elas- crystal are immobile and their interaction can be tic energy is equal to expressed by a pair central potential. Let one atom be the origin of coordinates. Stability condition for ddUc==σε ε d, ε ij ij ijkl kl ij crystal lattices affirms that in this case the elastic or in the integral form constants of mono-atomic cubic crystals have the form (Max Born, 1940) 1 Uc= εε. 4 ijkl kl ij a 2 c = ∑ mDUr42af, 11 2ν ii i The expression characterizes the specific work 0 i done by the stress under elastic deformation of a a4 cc== ∑ mnDUr22 2 af, medium (usually continuum), and for this reason 12 44 ii i i 2ν i ia named the elastic potential. 0 1 d The tensor of elastic constants has 81 compo- = Di . nents, but due to the symmetry of the strain tensor riidr the product ε ε does not change if to permute kl ij Here ν is the atomic volume, a is the lattice pa- indices i and j, k, and l, or pairs of indices (I,j) and 0 =++222 (k,l). Therefore rameter, rmnliiii is the distance between the origin of coordinates and atom i. Note === ccccijkl jikl ijlk klij . that the equality c12 = c44 is known as Cauchy’s As a result, the number of different components relation. reduces to 21. In crystals, by virtue of their high These formulas have played and are playing symmetry there appear additional relations between up to now an awful role in the development of mo- different components of the tensor of elastic con- lecular dynamics; to say more correctly, not the stants. As a consequence, the number of indepen- formulas but people using them. Indeed, the for- dent components decreases even more. For ex- mulas allow solve two types of problems: ample, cubic crystals have only three components 1) Knowing the interatomic potential to calculate which are labeled by two ways the elastic constants. This is a direct problem. 2) On the basis of experimental elastic constant === find parameters of the interatomic potential. This cccccc1111 11,,. 1122 12 1212 44 is an inverse problem. To clarify the physical sense of these linear-in- “Conditionally, the theoreticians dealing with dependent elastic constants, one can use three condensed matter physics can be separated into types of deformation parameters ν, ε , γ (K. Fuchs, i i two types. The first-type theoretician supposes that 1936). The first parameter ν characterizes homo- constants and functions which characterize solids geneous extension or compression of a crystal lat- are given from ‘above’, and it is necessary to find tice. Parameter ε describes the lattice extension 1 them experimentally. The activity of a theoretician in the direction x and the compression in the direc- consists in the following: to express observed physi- tion y in such a way that the crystal volume re- cal values through such constants and functions. mains constant. Parameter γ defines the y-axis 1 The second-type theoretician supposes that pre- rotation in the plane xy conserving a height, i.e. scribed values are not characteristics of solids but simple shear. The elastic constants suggested by those of atoms (and in general, only world con- Fuchs can be expressed as the partial derivatives stants), and the purpose of a theory is to calculate of elastic energy for an undeformed crystal with properties of solids on the basis of atom proper- respect to deformation parameters: ties, since the atoms constitute solids. Surely, this 2 ∂ U 1 is a very important problem. Its solution allows, in =+afcc2 – bulk or compressibility ∂ν 2 3 11 12 principle, predict the properties of solids which are modulus (all-side compression, volume changes), not obtained yet. There is no sense to compare ∂ 2U ∂ 2U merits and demerits of both approaches because =−2afcc, =c ∂ν 2 11 12 ∂γ 2 44 – shear moduli (vol- they are quite different and pose different problems. 1 ume does not change). It is clear that they are useful both” [6]. Potentials of interatomic interaction in molecular dynamics 11

Examples of the interatomic potentials for met- molecule parameters from the first principles [4], als developed on the basis of the first (macroscopic) and this approach is in accord with the modern ten- approach can be found in [7,8] and those on the dency in the development of condensed matter basis of the second (atomistic) one in [9,10]. Their physics. We will not discuss more the dead-end comparison is done in [5]. approach; instead we draw attention to the models It is worth adding that the second approach is of molecular dynamics which are based on the first more difficult, but it helps to clarify the nature of a principles. phenomenon in a more profound manner, and vast majority of success in physics is connected namely 6. NEW MOLECULAR DYNAMICS with the second approach. Unfortunately, because of large difficulties in realizing the second (atomis- 6.1. Born-Oppenheimer’s tic) approach, molecular dynamics community de- approximation in a new fashion velops this science mainly by the first more simple Consider the way of numerical calculation of an way, not bearing in mind that the first approach interatomic potential in which the potential is de- leads to a dead end many neighboring sciences. fined during the integration of motion equations (R. As a rule, Cauchy’s relation does not fulfill even for Car and M. Parrinelo, 1985). The model of mo- homogeneous solids, so one is compelled to intro- lecular dynamics using suchs approach was named duce many-body terms adjusting their parameters by the authors ab initio molecular dynamics In the to experimental data. The choice of parameters ab initio molecular dynamics one employs both the depends on the taste of an author; therefore, the atomic coordinates {Ri} and the electronic wave- number of empirical potentials even for one solid, ψ functions { i} which correspond to occupied states, in principle, is unlimited. For amorphous materi- as dynamical degrees of freedom of a fictitious clas- als, alloys and real metallic crystals with defects sical system governed by the following equations the situation transforms into a tragedy. In those of motion cases it is necessary to take into consideration local surrounding of an atom. This leads to infinite δE modifications and discussions about the choice µψ&& afr,t =− +∑ λψafr,,t i δψ * afr,t ij j correctness of one or other type of an empirical i j potential [11,12]. More detailed discussion needs && ∂E MR =− . a special review paper. II ∂R aft It should be emphasized that the more com- 1 plex is the structure of condensed matter, the more Here dots indicate time derivatives, M denote parameters one needs to describe its properties I atomic masses, µ is an adjustable parameter set- using the first (quasi-theoretical) approach [13]. At ting the time scale for the fictitious electronic dy- last, the number of parameters begins to exceed namics, E is the energy functional with the local the number of experimental data on the basis of density approximation and correlation, and λ are which it is possible to define the parameters; but ij Lagrangian multipliers. The Lagrangian multipliers ‘the value of the thing is as much as it will bring.’ are needed to satisfy the orthonormality constraints It is worth noting that a similar situation has on the ψ (r,t). When E is minimized with respect to appeared earlier in the theory of molecule vibra- i the ψ (r,t), one recovers the Born-Oppenheimer tions [14]. The corresponding problems are spo- i potential energy surface for atoms, and the sec- ken as the direct and inverse spectral ones. The ond equation describes physical atomic trajecto- direct spectral problem consists in calculating fre- ries. quencies and vibration spectrum on the basis of “With an appropriate choice of the parameters preset parameters of a molecule. The inverse spec- and the initial conditions, fast electronic and slow tral problem is reduced to the following: how to find atomic degrees of freedom are only weekly coupled the potential energy of a molecule on the basis of and atomic trajectories, initially lying on the Born- experimental frequencies. In the theory of molecule Oppenheimer surface, deviate from it very slowly vibrations the scientists have come many years ago on the time scale of the molecular dynamics simu- to the conclusion that the direct problem had a lation. In these conditions, the electronic degrees unique solution where the inverse one had in a of freedom acquire only very small classical kinetic general case no sole solution. For this reason, the energy needed to follow adiabatically the atoms, main efforts are directed now how to calculate and only very few or no separate minimizations are 12 A.I. Melker necessary to keep the system on the Born- electronic-atomic processes with a strong correla- Oppenheimer surface during a molecular dynam- tion and localization in space and time, particularly ics run. In this approach, potential is explicitly de- in studying fullerenes and nanotubes. The advan- rived from the electronic ground state, which is tage of the new approach is that the structure is treated with accurate density functional technique” not postulated a priori and, moreover, can undergo [15]. significant transformations with time. However, the In spite of such optimistic statement of the au- main advantage of the model is that it is offers con- thors, there are number of points to be made. siderable possibilities for analyzing the influence • The authors claim that the scheme is “param- of electric and magnetic fields on the fullerenes eter-free” and opens the possibility of predict- and nanotubes under dynamic conditions. ing from the first principles properties of disor- This new molecular dynamics was developed dered systems. However, we have shown that more than ten years ago initially with the purpose both the Born-Oppenheimer equations are ap- to study the inversion transition in an ammonia as proximate, so any solution for electrons depends well as positron scattering by atoms and molecules only on nuclei (atoms) coordinates, but not on (A.I. Melker, S.N. Romanov, A.L. Emiryan, 1996), their velocities. In its turn any solution of the and for a long time (up to 2007 [4]) had no rigorous equations for nuclei (atoms) depends only on theoretical grounds, although with its help many the total energy of electrons, but not on their interesting results on the self-organization and coordinates. properties of fullerenes and carbon nanotubes were • It seems strange that the first principles could be received; which cannot be obtained in the frame- approximate. Moreover, the scheme contains work of common as well as of ab initio molecular the adjustable parameter µ and at the same time dynamics. It should be emphasized that this ap- is “parameter-free”. In our opinion, the name proach is known under different names: charged- “ab initio molecular dynamics” is too ambitious. bond model, molecular dynamics model of ‘charges In reality, it is ab initio molecular “semi-dynam- at bonds’. This difference in names is due to trans- ics” because electrons take no real part in many lation from Russian to English and back. Now we processes being always the slaves of the ground assume that the term ‘bond–charge’ accounts bet- state. ter for physical sense, and will use it in future.

6.2. Outside of the Born-Oppenheimer 7. CONCLUSION approximation We have analyzed the principles lying at the basis In the molecular dynamics simulations of atomic of different interatomic potentials which are used systems considered above, the contribution of the in molecular dynamics studies. The main idea of electronic subsystem is taken into account by im- this paper is given in the introduction and will be plicitly assuming that the electrons are always in repeated here. ‘A person must know how the cal- the ground state. In order to avoid this disadvan- culations are going on. If one does not understand tage, we elaborated a molecular dynamics model how it is done, one sees only bare numbers, but of charges at bonds which take into account both their genuine significance is hidden in calculations’ atomic (ionic) and electronic degrees of freedom [2]. We are not going to argue against the allega- [16].The fundamental difference between our tion that ‘the electron theorist saw only the glaring model and other currently available models is that, inadequacies of empirical potentials, but was un- within classical mechanics, the bond charge model able to deliver the tractable models required to provides a means for investigating both atomic improve the situation’ [12]. Instead, it must be em- (ionic) and electronic subsystems simultaneously. phasized that including a lot of empirical param- This approach accounts for the contribution from eters into a molecular dynamics model often does the electronic subsystem even in the case when not clarify a subject, but immerses it into darkness. the electrons are in the excited state or when the To our mind, the main aim of the investigators work- system undergoes electronic transitions [1]. The ing in the field of molecular dynamics consists in proposed model makes it possible to observe the creating new and developing already existing mod- variation in the configuration on the electronic level els which are based on rigorous theoretical and to extend information thus derived to the atomic grounds. ‘A lot of talent is wasted to improve de- level closely related to the experiment. The model tails, whereas it is necessary to reconstruct all the is especially useful in investigating non-equilibrium Potentials of interatomic interaction in molecular dynamics 13 building as a whole’ (Henry Léon Lebesgue, 1875- [8] M.S. Daw and M.I. Baskes // Phys. Rev. B 29 1941). (1984) 6443. [9] A.A. Vasilyev, V.V. Sirotinkin and A.I. Melker // REFERENCES phys. stat. sol. (b) 131 (1985) 537. [10] A.I. Melker, D.B. Mizandrontzev and V.V. [1] A.I. Melker // Proceedings of SPIE 6597 Sirotinkin // Zeitschrift für Naturforschung (2007) 659701. 46a (1990) 233. [2] R.W. Hamming, Numerical Methods for [11] R.A. Johnson // Phys. Rev. B 37 (1988) Scientists and Engineers (Mc Graw-Hill Book 3924. Company, Inc. New York, San Francisco, [12] M.W. Finnis, A.T. Paxton, D.G. Pettifor, A.P. Toronto, London, 1962). Sutton and Y. Ohta // Phil. Mag. A 58 (1988) [3] A.S. Davydov, Quantum Mechanics (Nauka, 143. Moscow, 1973), In Russian. [13] Z.M. Frenkel and A.I. Melker // Proceedings [4] A.I. Melker and M.A. Vorobyeva // of SPIE 5127 (2003) 63. Proceedings of SPIE 6253 (2006) 625305. [14] M.V. Volkenstein, L.A. Gribov, M.A. [5] A.I. Melker, Modeling Experiment, Physics Elyashevich and B.I. Stepanov, Vibrations of Series No. 10 (Znanie, Moscow, 1991), In Molecules (Nauka, Moscow, 1972), In Russian. Russian. [6] A.L. Efros, From a translator, In: A.O. [15] F. Buda, G.L. Chiarotti, I. tich, R.Car and Animalu, Intermediate Quantum Theory of M. Parrinelo // J. Non-Cryst. Solids 114 Crystalline Solids (Prentice Hall,Inc., (1989) 7. Englewood Cliffs, New Jersey, 1977). [16] A.I. Melker, S.N. Romanov and A.L. Emiryan [7] A.N. Orlov and Yu.V. Trushin, Energies of // Pisma v Zh. Tekh. Fiz. 22 (1996) 65 (Tech. Point Defects in Metals (Energoatomizdat, Phys. Lett. 22 (1996) 806). Moscow, 1983), In Russian.