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Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica Introduction the Hartree Fo

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Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica Introduction the Hartree Fo European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom using Mathematica César R. Acosta*1 J. Alejandro Tapia1 César Cab1 1Av. Industrias no Contaminantes por anillo periférico Norte S/N Apdo Postal 150 Cordemex Physics Engineering Department Universidad Autónoma de Yucatán *[email protected] (Received: 20.12. 2013, Accepted: 13.02.2014) Abstract Slater type orbitals were used to construct the overlap and the Hamiltonian core matrices; we also found the values of the bi-electron repulsion integrals. The Hartree Fock Roothaan approximation process starts with setting an initial guess value for the elements of the density matrix; with these matrices we constructed the initial Fock matrix. The Mathematica software was used to program the matrix diagonalization process from the overlap and Hamiltonian core matrices and to make the recursion loop of the density matrix. Finally we obtained a value for the ground state energy of helium. Keywords: STO, Hartree Fock Roothaan approximation, spin orbital, ground state energy. Introduction The Hartree Fock method (Hartree, 1957) also known as Self Consistent Field (SCF) could be made with two types of spin orbital functions, the Slater Type Orbitals (STO) and the Gaussian Type Orbital (GTO), the election is made in function of the integrals of spin orbitals implicated. We used the STO to construct the wave function associated to the helium atom electrons. The application of the quantum mechanical theory was performed using a numerical method known as the finite element method, this gives the possibility to use the computer for programming the calculations quickly and efficiently (Becker, 1988; Brenner, 2008; Thomee, 2003; Reddy, 2004). It is known that the effects of electron spin produces an antisymmetric wave function and that the charge of the nuclei is screened by the other electrons, then the effective charge that affect the interactions electron-nuclei and electron-electron is ζ, all of this effects are considered on the Hartree Fock Roothaan approximation. Dealing with Restricted Closed shell approximations means that we are considering only an even number of electrons, with all electrons paired such that n=N/2 spatial orbitals are doubly occupied (Szabo, 1996; Levine, 2008; Bransden, 2003). Hartree Fock Roothaan equations The initial point of view of this approach is the independent particle model, according to which each electron moves in an effective potential which takes into account the attraction of the 1 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab nucleus and the average effect of the repulsive interaction due to the others electrons and their spin (Hartree, 1957; Rosetti, 1974; Szabo, 1996; Levine, 2008; Bransden, 2003). The general procedure for finding a wave function of order zero (ground state) of a set of electrons, is to multiply the hydrogen-like wave function of each one electron Where the hydrogen-like orbitals are m f= Rnl( rY) l (θϕ, ) m Where Rrnl ( ) is the radial function and Yl (θϕ, ) are spherical harmonics. This wave function has total nuclear charge Ze, with Z number of protons and e the charge of an electron. Each electron in a many electron system is described by its own wave function (Bransden, 2003). According with this approach the trial wave function of the N-electrons is a Slater determinant Φ, which is a function of spatial orbitals and spin states. The central modification made at this point is to assume that an electron in a multi-electron atom not perceive the whole nuclear charge, as this is partially shielded by the other electrons. Therefore it is proposed an antisymmetric wave function of the form (Szabo, 1996; Levine, 2008; Bransden, 2003) Where gn represent the spin orbital, but now depends of shielded nucleus charge ζ, which can be expressed as a Linear combination of Atomic Orbitals (LCAO), depending on the base functions ( ) as follows gcn= ∑ iiχ (1) i=1 Where the base function are a complete set of Slater Type Orbitals (STO), which is defined in atomic units as (2) in this STO n represents the principal quantum number of the electron orbital for which the function is defined, and ζ is an effective electron charge (Rosetti, 1974). The Hartree-Fock-Roothaan equations based on the STO functions are (3) 2 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab in equation (3) Frs represents a general element of the Fock Matrix, rs tu are the electronic repulsion integrals and Ptu are the elements of the density matrix, that are written as n/2 * Ptu =2∑ cctj uj ; t =…=… 1, 2, , b ; u 1, 2, , b (4) j=1 electronic repulsion due to the charge and spin of the electrons expressed in the spin orbital are given by the integrals (5) These integrals represent the expected value of the interaction 1/r12 relative to the state χχrs()ij ( ) such that the electron i is in spin-orbital χr and the electron j is in spin-orbital χs . The core H rs are the elements of the Hamiltonian operator that are given by core ˆ core HHrs= χχ r (11) s ( ) (6) Where the Hamiltonian operator written in function of the screened charge ζ is (7) The Roothaan eigenvalue equation is then expressed as b ∑csi( F rs−=⇒εε i S rs ) 00Determinant( Frs−= i S rs ) (8) s=1 Finally the elements of the overlapping matrix are the integrals given by (9) Integral evaluation There are two kinds of integrals, the one-electron as are the overlapping integral Srs and the core Hamiltonian core H rs , and the two-electron given by the electron repulsion integral rs tu , each one has different treatment. For the one-electron integrals, we used the formula shown below, that can be obtained integrating by parts ∞ b! xb e− qx dx = ∫ b+1 (10) 0 q 3 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab and for two-electron integral we used (10) and additionally these two ones xx2 22 x2 e−−bx dx=− e bx ++ ∫ 23 (11) b bb e−bx xe−bx dx=−+( bx 1) (12) ∫ b2 Since in the application of equation (5) to the spin orbital of two electrons recursively we have the integral that is shown below ∞∞−br2 − e = ar1 22 A∫∫ e r11 dr r 22 dr (13) 00r12 This integral can be transformed to an evaluation formula as is described next. We know that r12 represents the distance between the two electrons and that dr2 is the differential radial in the space of the electron 2, then we have ∞ − = ar1 2 A∫ e r11 dr B (14) 0 Where B is given by r1 ∞ 1 −− B= r2 ebr22 dr+ r ebr dr r ∫∫2 22 2 1 0 r1 Then applying (\ref{11}) to the first part of B and (12) to the second part, we have 2− 12 =−+br1 Be3 23 (15) br11 b br Substituting (15) in (14) and simplifying we obtain ∞∞ ∞ 21− −+(a br) 2−+(a br) A=−− ear1 r dr e11 r2 dr e r dr 32∫∫11 1 1 3 ∫11 (16) bb00 b 0 Using equation (10) to evaluate (16), finally obtain a formula for the electron repulsion integral, as 22 2 =−− A 32 3 2 (17) ba (abb++) 23( abb) 4 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab Therefore, if we want to calculate the ground state of helium atom with the SCF procedure, using the orbitals exponent ζ1 = 1.45 y ζ 2 = 2.91 (Roetti, 1974), the spin orbital functions are, according with equation (2) 3 2 −ζ1r 0 χζ11= 2 eY 0 (18) 3 2 −ζ 2r 0 χζ22= 2 eY 0 (19) Applying (5) to the state 1111 , we have Substituting spin orbital χ1 and simplifying 2 −2ζ12r 0 − ζ 2 eY( 0 ) = ζττ602 11r 1111 16 1∫∫e( Yd 01) d2 (20) r12 2 2 with dτ1= r 1 Sin θ 11 dr d θφ 1 d 1, dτ2= r 2 Sin θ 22 dr d θφ 2 d 2 and ππ2 2 0 θθ φ= ∫∫(Y0 ) sin dd 1 00 Equation (20) is written as ∞∞−2ζ12r − ζ e = ζ 622 11r 2 1111 16 1∫∫e r 11 dr r 22 dr 00r12 Then using (17) with ab= = 2ζ1 5 1111=ζ = 0.9062 (21) 8 1 for the state 22 22 , we have the same result, but with ζ 2 5 22 22=ζ = 1.8188 (22) 8 2 Again applying (5) to the state 12 12 , we have 5 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab χχχ**(112) ( ) ( ) χ( 2) = 1 21 2 ττ 12 12 ∫∫ dd12 r12 Substituting spin orbitals χ1 , χ2 and simplifying ∞∞−+(ζζ1 22)r −+(ζζ)r e = ζζ33 1 21 2 2 12 12 16 12∫∫e r11 dr r22 dr 00r12 with ab= =(ζζ12 + ) substituting in (17) and simplifying 20ζζ33 =12 = 12 12 5 0.9536 (23) (ζζ12+ ) Once again applying (5) to the state 11 22 , we have χχχ**(11) ( ) ( 2) χ( 2) = 112 2 ττ 11 22 ∫∫ dd12 r12 Substituting spin orbitals χ1 , χ2 and simplifying ∞∞−2ζ 22r − ζ e = ζζ332 11r 2 2 11 22 16 12∫∫e r 11 dr r 22 dr 00r12 with a = 2ζ1 and b = 2ζ 2 substituting in (17) and simplifying 11 1 11 22 =−−ζζ33 =1.1826 (24) 12ζζ23 3 232 12 (ζζζ122++) ( ζζ 12) ζ 2 Again applying (5) to the state 1112 , we have χχχ**(11) ( ) ( 2) χ( 2) = 1 11 2 ττ 1112 ∫∫ dd12 r12 Substituting spin orbitals χ1 , χ2 and simplifying 93∞∞−+(ζζ1 22)r − ζ e = ζζ22 2 11r 22 1112 16 12∫∫e r 11 dr r22 dr 00r12 with a = 2ζ1 and b =ζζ12 + substituting in (17) and simplifying 6 European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab 3 933ζζ+−+ 4 ζζζ22 −43 ζζζ + 22( 12) ( 121) ( 121) 1112= 16ζζ (25) 12 332 23( ζζ12++) ( ζζζ 121) it's easy to observe that 1112= 12 11 = 11 21 = 2111 = 0.9033 (26) Finally applying (5) to the state 12 22 , we have Substituting spin orbitals χ1 , χ2 and simplifying 39∞∞−2ζ 22r −+(ζζ)r e = ζζ22 221 21 12 22 16 12∫∫r 1 e dr1 r 2 dr 2 00r12 with a =ζζ12 + and b = 2ζ 2 substituting in (17) and simplifying 39 22 11 1 12 22= 16ζζ −− 12 32323 2 4ζζζ212( ++) 23( ζζζ 122) 43( ζζζ 122 +) it's easy to observe that 12 22= 22 12 = 21 22 = 22 21 = 1.2980 (27) The one-electron integrals are simpler than the two-electron integrals, then
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