# Hartree-Fock electronic structure calculations for free atoms and immersed atoms in an electron gas

T ABLE OF CONTENTS Page 1. INTRODUCTION...........................................................1 2.HARTREE-FOCK THEORY................................................4 2.1 The Many-Body Problem...............................................4 2.1.1 Free Electron Gas...............................................4 2.1.2 Hydrogen Atom.................................................5 2.1.3 Many Electron Atoms...........................................7 2.2 Basic Theory:Hartree-Fock............................................8 2.2.1 Assumptions,Approximations and the Fock operator............8 2.2.2 Total Fermion Wave Function:Slater Determinant..............10 2.2.3 Single Particle Contributions....................................11 2.2.4 Two Particle Contributions.....................................12 2.2.5 Variational Principle and the Schr¨odinger Equation..............13 2.2.6 Hartree-Fock Equations.........................................14 2.3 Basis State Expansion..................................................15 2.3.1 Radial Wave Functions..........................................17 2.3.1.1 Density Functional Theory.................................17 2.3.1.2 Electron Occupation.......................................20 2.3.2 Angular Wave Functions:Spherical Harmonics..................21 2.4 Numerical Implementation..............................................22 2.4.1 Kinetic Energy.................................................22 2.4.2 Nuclear Coulomb Energy.......................................23 2.4.3 e-e Coulomb Energy............................................23 2.4.4 Unrestricted Hartree-Fock.......................................25 2.4.5 Generalized Eigenvalue Problem................................26 2.4.6 Self-Consistent Field............................................27 2.4.7 Computational Details..........................................29 2.5 Post Hartree-Fock......................................................29 2.5.1 Conﬁguration Interaction.......................................31 2.5.2 Multi-Conﬁgurational Hartree-Fock.............................31

T ABLE OF CONTENTS (Continued) Page 3. SINGLE ATOM ELECTRONIC CONFIGURATIONS.......................33 3.1 Electronic Conﬁgurations...............................................33 3.2 Eigenvalues,Eigenvectors and Total Energy.............................35 3.2.1 Ground State Total Energy.....................................37 3.2.2 Helium.........................................................41 3.2.3 Lithium........................................................44 3.2.4 Beryllium.......................................................47 3.2.5 Boron..........................................................51 3.3 A Detailed Example:Carbon...........................................54 3.4 Basis Set Completeness.................................................62 4.EXPERIMENTAL VERIFICATION.........................................64 4.1 Addition of Angular Momentum........................................65 4.2 Many-Fermion Angular Coupling.......................................67 4.3 Results:Boron.........................................................69 5.IMMERSED ATOM.........................................................71 5.1 Theory Overview.......................................................71 5.1.1 Basis State Extensions..........................................72 5.1.2 L¨owdin.........................................................77 5.1.3 Hydrogen Function Perturbation................................79 5.2 Numerical Implementation:First Order.................................83 5.2.1 Immersed Kinetic Energy Terms................................83 5.2.2 Immersed Nuclear Energy Terms................................85 5.2.3 Direct and Exchange Terms.....................................86 5.2.4 k-space Integrations.............................................88 5.3 Immersed Self Consistent Field.........................................91 5.3.1 Immersed Energy Eigenvalues...................................91

T ABLE OF CONTENTS (Continued) Page 5.3.2 Maximum Number of States....................................92 5.3.3 Fermi Wave Vector:k f ..........................................93 5.4 Results.................................................................93 6.CONCLUSION..............................................................101 BIBLIOGRAPHY...............................................................103

LIST OF FIGURES Figure P age 2.1 Example of converged DFTstates used for the HF basis of carbon.Radial distances r,are measure in units of Bohr radii...........................19 2.2 Carbon 3s,3p and 3d basis states with diﬀerent fractional occupations. The fraction is expressed by ”exp 0.25”.................................21 2.3 Flow chart of iterative HF scheme.Immersed atoms have an extra loop to form the immersed Fock matrix from each states previous eigenvalue..28 2.4 The energy decreases as correlation eﬀects are accounted for.Exact so- lutions to the relativistic Dirac equation yields the lowest Hartree-Fock energies.................................................................30 3.1 Eigenvalues for oxygen in the ground state electron conﬁguration E[(2s);21− 1 ↑,210 ↑,211 ↑,21 −1 ↓] with nmax=3.The lowest 2p state is occupied by the 210 spin up electron,the middle 2p state is degenerate among the 21-1 and 211 spin up electrons..........................................36 3.2 Eigenvalue of the 1s spin-up electron for helium through oxygen.........40 3.3 The mostly occupied 1s and mostly unoccupied 2s and 3s eigenvector coeﬃcients for helium to ﬂuorine........................................41 3.4 Heliumeigenvalues for nmax=3 with a basis set generated with the lowest fractional occupation.Switching of eigenvalues is shown with the vertical lines of one of the sets.................................................43 3.5 Helium 1s eigenvector for various fractional charge excitations in DFT basis set generation.nmax=3..........................................44 3.6 Lithium total energy,during convergence,for diﬀerent basis sets (exp 0.2 to 1.0).................................................................46 3.7 Lithium eigenvalues for nmax=3 with a basis set generated with the lowest fractional occupation.............................................47 3.8 Beryllium total energy,during convergence,for diﬀerent basis sets (exp 0.2 to 1.0).............................................................49 3.9 Beryllium with nmax=3.2s spin orbit eigenvector for a number of frac- tionally excited basis sets..............................................50 3.10 Beryllium eigenvalues for nmax=3 with a basis set generated with the lowest fractional occupation............................................51

LIST OF FIGURES (Continued) Figure P age 3.11 Boron total energy,during convergence,for diﬀerent fractional excitation basis sets (exp 0.2 to 1.0)...............................................53 3.12 Boron eigenvalues for nmax=3 with a basis set generated with the lowest fractional occupation...................................................54 3.13 Total Hund’s rule predicted ground state energy of carbon during con- vergence for nmax=3 and 0.25 fractional excitation.....................57 3.14 Eigenvector of the 2p(m=-1) spin orbital of carbon during convergence. The basis states were generated with 0.25 fractional excitation and has nmax=3................................................................58 3.15 Total energy of an excited E[(1s);200 ↑,21 − 1 ↑,210 ↑,211 ↑] state of carbon during convergence instabilities.The basis states were generated with 0.25 fractional excitation and has nmax=3.........................59 3.16 2p spin-up degenerate eigenvalues lifted for an excited E[(1s);200 ↑,21− 1 ↑,210 ↑,211 ↑] state of carbon.The basis states were generated with 0.25 fractional excitation and has nmax=3..............................60 3.17 2p spin orbit eigenvector for an excited E[(1s);200 ↑,21−1 ↑,210 ↑,211 ↑ ],non-convergent,state of carbon.The basis states were generated with 0.25 fractional excitation and has nmax=3..............................61 5.1 1s basis state of carbon as it is dispersed away from the nucleus by R new (r) = R old (r)r cosh(c √ E(r −R)). ..................................74 5.2 Energy of carbon with basis states extended out from the nucleus with R new (r) = R old (r)r cosh(c √ E(r−R)). The electronic state is E[(2s);210 ↑ ;211 ↑]..................................................................75 5.3 2p eigenvector coeﬃcients for carbon with dispersed states.Electron conﬁguration is E[(2s);210 ↑;211 ↑].....................................76 5.4 1s hydrogen state coupled to two plane wave states with wave num- bers k=1 and kappa=2.First and second order L¨owdin perturbation is compared to the exact solution.The vertical axis is U 1 ,U 2 and the det[Hamiltonian]........................................................80 5.5 Two hydrogen states coupled to two plane wave states.First and sec- ond order L¨owdin perturbation is compared to the exact solution.The vertical axis is U 1 ,U 2 and the det[Hamiltonian].........................82

LIST OF FIGURES (Continued) Figure P age 5.6 Change total energy vs k f for helium through carbon immersed in an electron gas.Convergence was possible with nmax=2 and smax=2.......95 5.7 Data analysis of the k f dependence of the total energy of carbon im- mersed in an electron gas.Convergence was possible with nmax=2 and smax=2................................................................96 5.8 Immersion energy dependence on k f for helium through carbon.Each follow similar k f power dependence.Convergence was possible with nmax=2 and smax=2...................................................98 5.9 Immersed beryllium eigenvectors for nmax=2 and smax=2.The plot inset is to see the relative diﬀerences between the closely lying values in the top left plot.........................................................99

LIST OF TABLES Table P age 3.1 Total energy for many light atoms calculated,by direct and indirect treat- ments of the kinetic energy.These are compared to the exact,complete basis set,Roothaan-Hartree-Fock (RHF)energies.Basis set generated up to nmax=3.The energy diﬀerence is tabulated under indirect-Rooth.....39 3.2 Total energy of helium,for diﬀerent basis sets,compared to Roothaan HF.Basis refers to the fractional charge excitation in the DFT basis set generation.............................................................42 3.3 Total energy of lithium,for diﬀerent basis sets,compared to Roothaan HF.Basis refers to the fractional charge excitation in the DFT basis set generation.............................................................45 3.4 Total energy of beryllium,for diﬀerent basis sets,compared to Roothaan HF.Basis refers to the fractional charge excitation in the DFT basis set generation.............................................................48 3.5 Total energy of boron,for diﬀerent basis sets,compared to Roothaan HF.Basis refers to the fractional charge excitation in the DFT basis set generation.............................................................52 3.6 Total energy of carbon,for diﬀerent basis sets,compared to Roothaan HF.Basis refers to the fractional charge excitation in the DFT basis set generation.............................................................55 3.7 Total energy of carbon,for electron conﬁgurations with the same L,for diﬀerent basis sets......................................................56 3.8 Total energy of carbon with an electron excited into n=3 states.Broken symmetries allow for the large number of unique conﬁgurations..........62 3.9 Boron energies with diﬀerent basis sets and electronic conﬁgurations.The lowest energy conﬁguration is E[(2s);210 ↑] and the most complete basis is generated with excitations into the 3s state...........................63 4.1 Conﬁguration interaction energies for boron L,S,M l ,M s states.Com- parison to experiment for the ﬁrst excited state of boron diﬀers by milli- hartrees...............................................................70 5.1 Total energy for carbon,with various dispersions out from the nucleus with the equation R new (r) = R old (r)r cosh(c √ E(r − R)). This is com- pared to many diﬀerent basis states.....................................75 5.2 Immersed atom energies for helium through carbon with nmax=2 and smax=2.Immersions for k f beyond those reported were not possible....94

HAR TREE-FOCK ELECTRONIC STRUCTURE CALCULATIONS FOR FREE ATOMS AND IMMERSED ATOMS IN AN ELECTRON GAS 1.INTRODUCTION My advisor Dr.Jansen,early after agreeing to work with me,asked if I wanted to use tools to model atoms or create tools for modelling atoms.The work presented here is designed to lay the ground work for a new tool created to model atoms,both free and immersed in a gas of free electrons.The end goal of such a tool is to model atomic impurities in metals accurately. Previous work [1][2] has explored the approach of an atom in an electron gas in the context of Density Functional Theory (DFT).Spherical [3] and non-spherical [4] calcu- lations of this type have shown promise as fundamental tools.Density functional and Hartree-Fock (HF) theory are both used to approximate solutions to the many-body Schr¨odinger equation.The natural question is then,how would the immersed atomic system work if modelled with HF theory?With this goal in mind an alternative question quickly arose,would using a spherically symmetric converged DFT calculation,for the HF basis states,improve completeness?This work solves the many-body HF Schr¨odinger equation for free atoms and atoms immersed in an electron gas,using basis sets generated from DFT. The ﬁrst step in modelling an atom,immersed in an electron gas,is to model the atom itself.To provide complete control over electronic conﬁgurations we’ve broken both spatial and spin symmetries.This freedom to explore all electronic conﬁgurations

2 increases the value of this tool but also increases complexity.Interactions between states of the appropriate symmetry lift degeneracies in the energy levels of the atom.The ground state electronic conﬁgurations can then be tested if they obey Hund’s rules [5].Various basis sets can be generated by adjusting parameters in DFT.These basis sets are tested for completeness.Along with the kind of basis states,the number,or size of the basis set is also tested for completeness.I call this ”exploring the space”.This ﬂexibility lays a good foundation for future work.Computations performed here can be a foundation for more complicated calculations. A third question,the answer to which I felt necessary to validate the work,was how do the energies calculated by HF compare to spectroscopic experiments?The answer to this question resulted in a side project,to couple the orbital and spin angular momentum of many conﬁgurational state functions.Not as trivial a problemas presented in introductory texts [6][7],this required expressing a completely anti-symmetric coupled L and S state in terms of the uncoupled l,mand s,m s states.For many electrons this becomes tedious and an algorithm to account for all cases had to be formed.The end result is a conﬁguration interaction comparison to spectroscopic data that ﬁts nicely. Finally,the immersion into an electron gas could be implemented.The idea is to couple the free atom to a jellium background.Metals have many free electrons but maintain a net neutral charge.To achieve this a static uniform background positive charge will oﬀset the free electron negative charge.Observing how the atoms electronic states shift while increasing the coupling with the free electrons will explain the eﬀects of an impurity atom in a metal.A brief exploration into spreading out the basis states attempted to simulate the eﬀects of an immersion.This didn’t increase the degrees of freedom of the system as it doesn’t actually couple any bound states to any other states. As a consequence,it simply increased the total energy.To successfully lower the energy, state space had to be increased.Unfortunately HF doesn’t take kindly to large state

3 space, as the computation time increases like M 4 ,where M is the number of basis states. The method must account for interactions of a set of free electron states on a set of bound electron states,without simply adding the spaces together. A perturbation technique posed by L¨owdin [8] presented a solution to the problem of coupling two slightly interacting systems.The method treats the inﬂuence of the free electrons as a small perturbation on the bound electrons and folds that interaction back into the space of just the bound electrons.This technique is applied under the iterative context of HF and the ﬁnal converged atom is inﬂuenced by the electron gas.Through multiple function expansions the immersed Fock matrix is derived.A Bessel function expansion [9] [10] limits the maximum density of free electrons that can be solved.For those densities in which the system can be solved,the feature of lowering the total energy of an atom immersed in an electron gas is shown.The model even appears to distinguish between larger energy advantages to immersion,for those elements that are metallic. To forecast what the model would predict for greater electron densities,the immersion dependency on the change in total energy is extrapolated. The work presented here not only produces results for comparison but also sets the stage for further research.A more complete set of heavier free atoms modelled,with their spectroscopic comparisons,could warrant interest.Improving the immersions to include larger densities and generating more a thorough list of converged atoms would also be of interest.In reality this work is the launching point for many,more complicated systems, fromﬁeld dependencies to impurity clusters,to full band structure calculations of all these systems.We have built a fundamental tool to be used to model sophisticated many-body systems.

4 2. HARTREE-FOCK THEORY The techniques shown in this work are based on those posed by D.Hartree and V. Fock [11][12][13].The methods approximately solve the many-body Schr¨odinger equation. All derivations use the natural Hartree atomic units where e = m e = = 1/4π 0 = 1. The unit for energy,the hartree,equals two Rydbergs.Lengths are measured in units of bohr radii. 2.1 The Many-Body Problem Solving the problem of multiple bodies,mutually interacting,is of great interest in physics.The problem can be explicitly solved for one and two particle systems.For systems of three particles and greater (many),no closed form analytical solution exists. The best hope is to make appropriate assumptions and approximations to come near a correct solution.Diﬀerent techniques work well in approximating a given problem. Determining which problem solving scheme should be used depends on the nature of the question. Electronic interactions arise from diﬀerent forces than gravitationally attracted ce- lestial bodies,but the many-body problem persists on all scales.At the atomic scale,the equation developed by Schr¨odinger to describe quantum phenomena must be satisﬁed. This work models the atomic world and at the core is a method to approximate solutions to the many-body Schr¨odinger equation. 2.1.1 Free Electron Gas Before modelling particle systems it helps to describe a single particle.Newton described macroscopic bodies like baseballs and planets but for accurate descriptions of

5 microscopic bodies on the atomic scale,quantum descriptions are required.General static quantum systems satisfy the time independent Schr¨odinger equation 2.1.

HΨ(r) = EΨ(r) (2.1)

H is the Hamiltonian operator which performs operations on a wave function Ψ(r), to extract information about the energy of a system.The connection to experiment is that this wave function,when multiplied by its complex conjugate Ψ ∗ (r)Ψ(r),is equal to the probability amplitude of ﬁnding the particle.This probability amplitude must satisfy:

Ψ ∗ (r)Ψ(r)dr = 1 (2.2) This says you have a one hundred percent chance of ﬁnding the particle,somewhere in space. Since the potential energy of a free particle is zero the only term in the Hamiltonian is the kinetic energy,

T.

H =

T = − −→ ∇ 2 2m (2.3) Exact solutions to the problemof a free particle with kinetic energy exist [6].The functions that satisfy this problem are plane waves,and the energies associated with these plane waves are those of a free particle. Ψ k (r) = 1 √ V e ±i − → k · −→ r ,E k =

2 k 2 2m (2.4) Finite volumes have a discrete set of allowed k wave vectors due to ﬁnite space requiring the solutions go to zero at the boundary.As the volume gets larger the k values become more continuous. 2.1.2 Hydrogen Atom The least complicated system involving the interaction between bodies is that of two particles attracted to each other.This can be thought of as a single particle attracted

6 to a central potential created by a second particle.This is the case for the hydrogen atom. Since the gravitational force between an electron and a proton is 39 orders of magnitude smaller than the Coulomb force,gravity will be neglected.For macroscopic bodies with no net charge,gravity is the dominant force,but the shape of the gravitational potential is the same as that of the Coulomb potential.The mass of a proton is roughly 1836 times that of an electron.For this reason the motion of nuclei can be neglected and a static coordinate system can be used with the origin at the nucleus.With these assumptions the problem is that of an electron in a central Coulomb potential created by a static nucleus. The nuclear Coulomb potential operator

V n ,is negative the inverse of the distance between the two charges with e 2 =1.

V n = − 1 | − → r | (2.5) With the kinetic energy operator for a single particle and the potential operator above, the Schr¨odinger equation for the simple hydrogen atom can be written as follows.

H Ψ( r ) = (

T +

V n )Ψ( r ) = − −→ ∇ 2 2m Ψ(

r ) − Z r Ψ(

r ) = E Ψ( r ) (2.6) The discrete set of solutions to this equation are exact and can be readily solved [13].The normalized wave functions contain information about the probability of where to ﬁnd the electron under the inﬂuence of the central Coulomb potential. Ψ nlm (r,θ,φ) =

2 na 0

3 (n −l −1)! 2n(n +l)! e − ρ 2 ρ l L 2l+1 n−l−1 (ρ) · Y l m (θ,φ) (2.7) In the above equation ρ = 2r na 0 and a 0 is the Bohr radius equal to 4π 0

2 m e e 2 . The functions that describe the probability density in the radial direction are the generalized Laguerre polynomials L 2l+1 n−l−1 (ρ).Angular information about an electron in a hydrogen atom comes from the spherical harmonics Y lm (θ,φ).The discrete solutions are identiﬁed by a set of integers called the quantum numbers.n is the principal quantum number,l is the angular momentumquantumnumber and mis the magnetic quantumnumber.The allowed values of the hydrogen quantum numbers are:n = 1,2,3...∞;l = 0,1,...,n −1;m= −l,...,l.

7 2.1.3 Many Electron Atoms One and two particle systems are important because they constitute the only systems that are analytically solvable.This is important but most interesting problems involve more than two bodies.For many-body systems there is no closed form analytical solution. If the assumptions of the hydrogen atom persist and other electrons are added,new operators arise in the equations.The Coulomb repulsion between the i th and j th electron has the following associated electron-electron Coulomb operator.

V ee = 1 | − → r i − −→ r j | (2.8) Writing down the Schr¨odinger equation for each particle yields a problem that no math has solved:a coupled set of second order diﬀerential equations.Coupling occurs through the electron-electron Coulomb potential.The consequence of this coupling is that the Coulomb energy of the i th electron depends on the position of the j th electron and vice versa.Consider the many-body Schr¨odinger equation with the assumption of a total wave function comprised of separable single particle states ϕ [13].

H i +

H j + 1 | − → r i − −→ r j |

ϕ i ( −→ r i )ϕ j ( −→ r j ) = Eϕ i ( −→ r i )ϕ j ( −→ r j ) (2.9) The equation for the energy of the i th particle would be:

H i ϕ i ( −→ r i ) +v j ϕ i ( −→ r i ) = Eϕ i ( −→ r i ) −

ϕ j ( −→ r j )H j ϕ j ( −→ r j )

ϕ i ( −→ r i ) (2.10) with v j =

|ϕ j (r j )| 2 | − → r i − −→ r j | d −→ r j .(2.11) The equation for the j th particle would be the same with each index changed.These are hopelessly coupled equations,through the Coulomb potential v j .Closed form analytical solutions to these equations do not exist.The best hope is to make approximations and then solve the system numerically.All many-body problems are solved approximately.

8 The precision of the solutions then depends heavily on how your assumptions and approx- imations are chosen.This point is especially important with regards to computational time and desired precision. 2.2 Basic Theory:Hartree-Fock The theory used to approximate the many-body electron problemwas ﬁrst developed by Douglas Hartree.The technique was posed as a way to ﬁnd many-body wave functions by solving N-coupled equations for N-spin orbitals.John Slater [14] and Vladimir Fock independently realized the method didn’t account for the antisymmetric nature of fermions and the exchange energy associated with them.This method,coined Hartree-Fock theory (HF),could then be used to solve quantum systems of electrons.Systems that can take advantage of approximate solutions to the electronic Schr¨odinger equation include nuclear, atomic,molecular and solid state.In all of these systems the basic principles behind Hartree-Fock hold true but diﬀerent assumptions and approximations may be made. 2.2.1 Assumptions,Approximations and the Fock operator To set the ground work needed to derive the Hartree-Fock equations a number of assumptions and approximations must be made [13].These are speciﬁc to the problem of a single,stationary,non-relativistic atom. Born-Oppenheimer approximation - The total wave function can be separated into two parts. Ψ total = ψ electronic ⊗ψ nuclear (2.12) This can be justiﬁed because of the mass mismatch between the electrons and nuclei. The motion of the nuclei have little dependence on that of the electrons.Since this work only models single atoms,mutual interaction of nuclei,is not needed.A result of this assumption is zero kinetic energy of the nucleus.

9 gr avity - The large disparity between Coulomb and gravitational forces justiﬁes ignoring all gravitational eﬀects. non-relativistic - The atoms modelled are light enough (Z<36) to ignore relativistic eﬀects in the momentum operator.(reference something) central potential - The nuclear potential is assumed to be a central potential.This fact justiﬁes the use of spherical harmonics for the angular portion of the wave functions. Interactions with other electrons will cause the overall potential to not be central. spin orbitals - The spin orbital ψ i ( −→ r i ;σ i ) represents the total state of an electron. Each spin orbital is assumed to be orthogonal and can be separated into a spacial function φ i ( −→ r i ) and a spin state χ(σ i ). ψ i ( −→ r i ;σ i ) = φ i ( −→ r i ) ⊗χ(σ i ) (2.13) basis set - Each spin orbital is expanded into a set of wave functions called the basis set.The variational method is used to minimize the energy and form a generalized eigenvalue problem.Solutions to this problem require a linear combination of ﬁnite,not necessarily orthogonal,basis sets.The features of the basis set will be varied to explore its precision. spin dependence - Since relativistic eﬀects are ignored the electron spin is added ad hoc.Eﬀects that are spin-dependent such as spin-orbit or spin-spin coupling must be added as corrections after the electronic Schr¨odinger equations have been solved. Fermi-Dirac statistics - The total wave function must always be antisymmetric under the exchange of particles.Energy eigenfunctions are then assumed to be determined through a single Slater determinant of single particle wave functions. electron correlation - The correlation energy is the diﬀerence in energy between the exact solutions to the non-relativistic many-body Schr¨odinger equation and that of HF. E correlation = E exact −E HF (2.14)

10 The interaction of electronic ensembles is only taken into account ad-hoc.This is one of the shortfalls of standard Hartree-Fock theory. electron exchange - When using the Slater determinant to anti-symmetrize the total wave function,electron exchange eﬀects are completely accounted for.This is one of the beneﬁts of Hartree-Fock theory. With these assumptions the total Hamiltonian operator (

H) is comprised of a kinetic (

T),a nuclear Coulomb (

V n ) and an e-e Coulomb (

V ee ) operator for every electron in the system.

H =

T +

V n +

V ee = − N

i=1 −→ ∇ 2 i 2m − N

i=1 Z r i + N

i=1 N

j >i 1 | − → r i − −→ r j | (2.15) N is the total number of electrons.The e-e interaction sums over every particle’s inter- action with every other particle. 2.2.2 Total Fermion Wave Function:Slater Determinant For a single particle,or a electron in a central potential created by static proton,a single wave function is all that is required to describe the system.It would seemreasonable then to form the total wave function of a system of particles as a product of individual wave functions. Ψ = N

α=1 ψ α ( −→ r α ;σ α ) = ψ 1 ( −→ r 1 ;σ 1 )ψ 2 ( −→ r 2 ;σ 2 )...ψ N ( −→ r N ;σ N ) (2.16) Here σ α is a spin index for the α th particle.The problem with this assumption is that electrons are among a class of particles called fermions.Two fermions cannot be in the same state and the same location,at the same time.This means any wave function describing a system of such particles must be antisymmetric under exchange of particles. To satisfy this requirement Fock and Slater implemented a determinant on the location

11 of the particles [14]. Ψ = 1 √ N!

ψ 1 ( − → r 1 ;σ 1 ) ψ 1 ( −→ r 2 ;σ 2 )...ψ 1 ( −→ r N ;σ N ) ψ 2 ( −→ r 1 ;σ 1 ) ψ 2 ( −→ r 2 ;σ 2 )...ψ 2 ( −→ r N ;σ N ) . . . . . . . . . . . . ψ N ( −→ r 1 ;σ 1 ) ψ N ( −→ r 2 ;σ 2 )...ψ N ( −→ r N ;σ N )

(2.17) Here the 1 √ N! is a normalization factor due to the increased number of wave functions.Ap- plying this procedure ensures Ψwill be antisymmetric under exchange.A more convenient notation for the total wave function uses the antisymmetrizing operator

A. Ψ =

A[ψ 1 ( −→ r 1 ;σ 1 )ψ 2 ( −→ r 2 ;σ 2 )...ψ N ( −→ r N ;σ N )] =

A N

α=1 ψ α ( −→ r α ;σ α ) (2.18) with,

A = 1 √ N!

p (−1) p

P = 1 √ N!

1 −

ij

P ij +

P ij k −· · ·

(2.19) where

P is the permutation operator.

P ij permutes the coordinates of electron i and electron j.If an even number of permutations occurs the term is positive and if odd,the term is negative.The total energy of a system of fermions can then be determined using the Schr¨odinger equation. ε =

p (−1) p

N

α=1 ψ α ( −→ r α ;σ α )

H

P N

β=1 ψ β ( −→ r β ;σ β )

(2.20) 2.2.3 Single Particle Contributions When using a properly anti-symmetrized wave function a large number of combina- tions of terms arise.The contribution to the total energy from the kinetic T,and nuclear Coulomb V NE ,energies would appear to have many combinations. (T +V NE ) =