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Quantum Chemistry of Solids / 14-Hartree-Fock LCAO Method for Periodic Systems

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4

Hartree–Fock LCAO Method for Periodic Systems

4.1 One-electron Approximation for Crystals

4.1.1 One-electron and One-determinant Approximations for Molecules and Crystals

The complex quantum-mechanical systems that are molecules and crystals, consisting of a great number of atomic nuclei and electrons, can in many cases be modeled by the nonrelativistic Hamiltonian having the form

ˆ 1

2Ne

1

Na

1

 

 

 

 

i

 

 

 

 

 

 

 

H = 2

ri 2

Mj Rj + V (r, R)

(4.1)

=1

j=1

 

 

 

 

 

 

 

 

where ri are the coordinates of electrons (i = 1, 2, ..., 2Ne), Rj are the coordinates of nuclei (j = 1, 2, ..., Na) and V (r, R) is the energy of the Coulomb interaction of electrons and nuclei:

 

|

 

|

 

 

| −

|

| −

|

 

V (r, R) =

 

Zj Zj

 

+

 

 

1

 

 

Zj

 

(4.2)

 

Rj

 

Rj

 

i<i

ri

 

ri

ij

ri Rj

 

j<j

 

 

 

 

 

 

 

 

 

 

 

 

 

The Hamiltonian (4.1) is approximate as it does not take into account the spinorbit interaction and other relativistic e ects. The calculation of eigenfunctions and eigenvalues of the operator (4.1), i.e. the solution of the time-independent Schrodinger equation

ˆ

(4.3)

HΦ =

is possible only after applying some approximations. The first of them is the adiabatic approximation. It permits the motion of electrons and nuclei to be considered separately and is based on the large di erence in electron and nuclear masses (me << Mj ).

In the adiabatic approximation, first the problem of electronic motion is solved for fixed positions of nuclei

"#

1

ri

+ V (r, R) ψ(r, R) = W (R)ψ(r, R)

(4.4)

2

i=1

106 4 Hartree–Fock LCAO Method for Periodic Systems

The wavefunctions ψ(r, R) and the eigenvalues W (R) in (4.4) depend on the nuclear coordinates R as parameters. Then, the found eigenvalues W (R) are used as the operators of potential energy in the equation determining the nuclear motion:

 

1

 

j

1

 

(4.5)

 

 

 

 

2

=1

Mj Rj + W (R) χ(R) = εχ(R)

 

 

 

 

 

 

 

This way of solving (4.3) is equivalent to the representation of the wavefunction Φ in the form of the product

Φ(r, R) = ψ(r, R)χ(R)

(4.6)

Further corrections of this reasonable approximation may be obtained from adiabatic perturbation theory by using, as the small parameter, the value ( M1 )1/4 where M is the average mass of the nuclei. Equation (4.4) is often considered as an independent problem without any relation to the more general problem (4.3). This is motivated by the following reasoning. If the temperature is not very high, the nuclei vibrate about some equilibrium positions R(0). Thus, in calculating the electronic structure, only the configuraton with the nuclei fixed at their equilibrium positions R(0) is considered. The latter are typically known from experimental data (e.g., from X-ray or neutron-scattering crystallographic data). This means of electronic-structure calculation without using any other experimental data (except the equilibrium positions of nuclei) is often considered as made from first-principles. Often, the first-principle calculations are made with the geometry optimization when the positions of nuclei are found from the total-energy minimization. Formally, the results of such calculations correspond to zeroth temperature. We write the equation for the electronic function of the system in the form

 

 

 

 

 

 

 

|

 

|

 

 

 

 

 

 

 

Hˆ eψ =

Hˆ0(ri) +

 

 

 

 

1

ψ =

 

 

(4.7)

 

i,i

 

ri

ri

 

 

 

 

 

i

 

 

 

 

 

 

 

 

 

 

 

 

 

where

 

1

 

 

| −

 

j |

 

| j

 

j |

 

ˆ

(ri) =

 

 

 

 

Zj

 

 

+

 

Zj Zj

 

 

(4.8)

H0

 

ri

 

 

 

 

 

 

 

 

 

2

r

i

 

R(0)

 

R(0)

 

R(0)

 

 

 

 

j

 

 

 

 

 

j<j

 

 

 

 

 

 

We assume here that there is a finite number of electrons in the system. This is certainly true for molecules, but for crystals it implies that we are using the model of a finite but boundless crystal (cyclic model), i.e. we consider the bulk of a crystal with

ˆ

cyclic boundary conditions imposed on opposite sides. The Hamiltonian He, being an operator acting on the functions depending on the electron coordinates ri is invariant under symmetry operations transforming the nuclear equilibrium configuration into itself. Later, we call it the symmetry group of a crystal.

Equation (4.7) is still very complex. Its solution is not yet possible without further simplifying approximations. The variables (electron coordinates) cannot be separated in (4.7) because of the terms describing the mutual Coulomb repulsion between electrons. This is why the exact many-electron wavefunction ψ may not be represented as a product (or the finite sum of products) of one-electron functions. However, the

˜

approximate wavefunction ψ may be taken as a sum of the products of one-electron

4.1 One-electron Approximation for Crystals

107

˜

functions, which have to be chosen so that the approximate function ψ is as close as possible to the exact solution ψ.

ˆ

The Hamiltonian He does not contain the spin operators. Therefore the oneelectron functions may be expressed in the form of the product ϕ(ri)α±(σi) of functions ϕ(ri) depending on the spatial coordinates of electrons and functions α±(σi) depending on spin variables only (α, β notations are also used for α±(σi)). The function ψ has to be antisymmetric under the exchange of any pair of electrons. This requirement is satisfied by the antisymmetric product of one-electron functions

ψ = [(2Ne)!]1/2 det [ϕ1(r1)α+(σ1) . . . ϕNe (rNe )α(σNe )]

(4.9)

The function ψ has to transform according to one of the irreps of the symmetry group G (point group for molecules or space group for crystals) We restrict ourselves to calculation of the ground state of the system. The latter is assumed to be invariant under the elements of the symmetry group G and to correspond to a total spin equal to zero. In this case, the wavefunction may be written in the form of a single determinant (one-determinant approximation).

In the Hartree–Fock method (also known as the one-electron approximation or self-consistent field (SCF) method) the system of equations for one-electron functions is obtained from the variational principle minimizing the functional

 

 

E = ψ Hˆ eψdτ

(4.10)

provided

 

 

 

 

ϕi (r)ϕi (r)dr = δii

(4.11)

The supplementary conditions (4.11) are not the primary restriction of generality. Indeed, the orthogonalization procedure of one-electron functions may be fulfilled directly in the determinant (4.9) without any change in its value. The same condition (4.11) assures the normalization of the function to 1. The one-electron functions satisfy the following system of Hartree–Fock equations:

ϕi(r) = i

εi iϕi (r)

(4.12)

or

 

 

 

 

ˆ

= ϕiε

(4.13)

Fϕi

where the Hartree–Fock operator

 

 

 

 

ˆ ˆ

 

ˆ

ˆ

(4.14)

F = H0

+ J

K

is defined below, ϕ is a row of functions ϕi and ε is the matrix of coe cients εi i.

ˆ

ˆ

is the one-electron operator (4.8).

In the Hartree–Fock operator F the term H0

ˆ

The action on the function ϕi(r) of the Coulomb operator J and exchange operator

ˆ

K is determined in the following way. Denote by ρ(r, r ) the mixed electron density with fixed spin (spinless electron-density matrix)

Ne

 

i

 

ρ(r, r ) = ϕi(r)ϕi (r ) = ϕ(r)ϕ

(4.15)

=1

 

108

4 Hartree–Fock LCAO Method for Periodic Systems

 

where ϕis the column of functions ϕi (r ). Then

 

 

 

 

 

(r , r )

 

 

Jˆϕi(r) = 2

 

ρ

 

dr ϕi(r)

(4.16)

 

|r − r

|

 

Kˆ ϕi(r) =

ρ(r, r )

 

 

 

 

 

ϕi(r )dr

(4.17)

 

|r − r |

ˆ

The operator J is the potential energy operator of an electron in the Coulomb field created by all the electrons of the system, with the electron in question included. This is a normal multiplication operator. It acts on the functions depending on the space coordinates of an electron. The Coulomb operator is determined if the electron

ˆ

density ρ(r , r ) is known. The operator K is an integral nonlocal operator with the

|

r

|

. To calculate the function Kˆ ϕ

i

ˆ

nucleus ρ(r, r )/

 

r

 

it is necessary to know the

function values in the entire domain of its determination. The operator F (4.14) is self-adjoint, as ρ(r, r ) = ρ (r , r).

Multiplying both sides of (4.12) by ϕj (r), integrating over dr and using the or-

thonormality of the functions ϕi (r), we obtain

 

εji = ϕj Fˆϕidr = ϕj Fˆϕidr = εij

(4.18)

The matrix ε in (4.18) is Hermitian. It may be diagonalized with the help of a unitary transformaton U:

F(ˆ ϕU) = ϕεU = (ϕU)(U1εU),

(4.19)

where U 1εU is a diagonal matrix. The unitary transformation U (UU= 1) does not change the electron-density matrix

ρ(r, r ) = ϕ(r)ϕ(r ) = ϕUUϕ= (ϕU)(ϕU)

(4.20)

ˆ

ˆ

 

Then, this transformation keeps the Coulomb J and exchange K operators invariant

ˆ

 

(and the total F). Therefore, without any restriction on the generality the matrix ε

in the right-hand side of (4.13) may be considered as diagonal

 

ˆ

(4.21)

Fϕi = εiϕi

The system of Hartree–Fock equations (4.21) is nonlinear. To solve it, an iterative method is usually used. In the course of the pth iteration the electron-density matrix

(p) ˆ ˆ

ρ (r, r ) and hence the operators J and K are considered to be fixed. The system (4.21) then transforms into one linear equation with a fixed self-adjoint operator

ˆ(p)

ˆ

ˆ(p)

ˆ

(p)

(4.22)

F

= H0

+ J

K

 

The eigenstates of this operator corresponding to the lower eigenvalues are populated with electrons. The occupied states are used to construct ρ(p+1)(r, r ) according to

ˆ(p+1)

ˆ

(p+1)

according to (4.16) and (4.17). Then, the subsequent

(4.20) and then J

and K

 

iteration is performed. The procedure of solution is stopped when the functions ϕ(jp) approach ϕ(jp−1)(r) (within the desired accuracy). In practice, other criteria are also

4.1 One-electron Approximation for Crystals

109

used to estimate the accuracy of the obtained solution (total energy, electron-density convergence criteria).

Thus, by an iterative method a self-consistent solution of the nonlinear equations (4.21) may be obtained. The rate of convergence of the iterative procedure depends on the nature of the physical system as well as on the choice of the initial approximation.

ˆ

The calculation of the eigenfunctions of the operator F during the pth iteration is itself a di cult problem that may be solved only approximately. To simplify it, the

ˆ

nonlocal exchange potential K is often replaced by a local potential. The one possible form of the local exchange potential is (especially for crystals)

ˆ

1/3

ϕi

(4.23)

Kϕi = 3α (3/4π)ρ(r, r)

 

proposed by Slater. The constant α in crystal electronic-structure calculations is an adjustable parameter. The widely used local density approximation (LDA) is considered in Chap. 7. If no iterative procedure is performed the electron structure obtained is said to be nonself-consistent. This kind of calculation is justified only when

ˆ

the one-electron Hamiltonian F(0) correctly reflects the main features of the exact (self-consistent) energy operator. It is very di cult to guess such a potential without involving some empirical data. The sum of atomic (or ionic) potentials is the simplest and most natural form of the molecular or crystal potential in nonself-consistent calculations. It is also used as an initial approximation in self-consistent electronicstructure calculations. The one-electron Hamiltonian approximation using the LCAO form of crystalline orbitals is known as the tight-binding (TB) method. This nonselfconsistent approach was popular in the 1970s when the self-consistent Hartree–Fock calculations were possible only for simple crystals. At the present time the TB scheme is mainly used as the interpolation scheme: LCAO Hamiltonian parameters are found by fitting to the band structure calculated self-consistently, for example, using a planewaves basis. Comparing calculation schemes of molecular quantum chemistry with those of quantum chemistry of solids one can say that the TB scheme is a semiempirical extended H¨uckel theory with periodical boundary conditions. The considered one-electron and one-determinant approximations are applied both for molecules and crystals. The symmetry of the one-electron Hamiltonian in both cases coincides with the symmetry of the nuclei configuration only for the case of closed-shell molecules or nonconducting crystalline solids. This is shown in next section.

4.1.2 Symmetry of the One-electron Approximation Hamiltonian

ˆ

The many-electron Hamiltonian He in (4.7), acting in the space of functions depending on the coordinates of all the electrons, is invariant under the operators gˆ of the equilibrium nuclear configuration symmetry group G. The operators of electron kinetic energy in (4.8) and the Coulomb repulsion between electrons in (4.7) remain unchanged under any transformation belonging to the pointor space-group operations since, according to the definition, the elements of theses groups are the transformations that do not change the distances between any two points of the space. As to the operators of electron–nuclear interaction in (4.8), the operations gˆ permute the members of the sum, leaving the sum as a whole invariant. Indeed, the coordinate ri transforms under g in

110 4 Hartree–Fock LCAO Method for Periodic Systems

ri = gri

(4.24)

and the denominator in (4.8) becomes equal to

 

|ri − Rj(0)| = |gri − Rj(0)| = |g(ri − g1Rj(0))| = |ri − g1Rj(0)|

(4.25)

i.e. to the denominator of other members in the sum in (4.8). This reasoning also

ˆ

proves the invariance of the one-electron operator H0 (4.8) under the transformations g from the group G.

The set of one-electron functions transforming according to the nβ –dimensional irrep D(β) is called the shell. For molecules, these shells are connected with irreps of the point-symmetry group. For a crystal, β = ( k, γ) – full irreducible representation of space group G, defined by the star of wavevector k and irrep γ of the point group of this vector. Taking into consideration the spin states α±(σ) we have 2nβ one-electron states in the shell. The functions ϕ(iβ)(r)α±(σ) span the space of the rep D(β) × D1/2, where D1/2 is the rep according to which the spin functions transform. For the systems with closed shells (molecules in the ground state and nonconducting crystals) the determinant (4.9) consisting of the functions of filled shells describes the state invariant under the operations of the group G (transforms over unity irrep of G). In fact any transformation of G replaces the columns of the determinant by their linear combinations, which does not change the value of the determinant, i.e. does not change the many-electron wave function ψ. In the case of open-shell systems (molecular radicals or metallic solids) the ground state of the system is described by a linear combination of determinants, appearing under the symmetrization over the identity irrep of group G. For crystals, the one-determinant approximation is practically the only way to make the electronic-structure calculations.

Now we check that for the closed-shell systems the group G is a symmetry group of the Hartree–Fock equations (4.21). First we note that the sum

nβ

 

 

 

i

(r )] ϕ(β)

 

 

[ϕ(β)

(r)

(4.26)

i

i

 

 

=1

 

 

 

is invariant under the group G, i.e. transforms over identity irrep of G. This follows

from the well-known property for finite groups of the direct product of irreps D(β) ×

 

 

. The electron-density matrix ρ(r, r ) is also invariant under G because for

D(β)

 

filled shells it is the sum of terms like (4.26)

 

 

 

 

 

 

 

 

 

ρ(gr, gr ) = ρ(r, r )

(4.27)

 

 

ˆ

 

 

ˆ(p)

, (4.22) are one-electron, i.e. they are determined

The operators F (4.21) and F

in the space of functions depending on the coordinates of one electron.

 

 

 

 

 

 

ˆ

 

 

 

ˆ

The invariance of the operator H is already proved. For Coulomb J and exchange

ˆ

 

 

 

 

 

 

 

 

 

 

 

K operators and for an arbitrary function ϕ(r) we have

 

 

 

gˆJˆϕ(r) = 2

 

|g1r

,− r |dr ϕ(g1r)

 

 

 

 

 

 

 

 

 

ρ(r

r )

 

 

 

 

= 2

ρ(g1r , g1r )

d(g1r )ϕ(g1r)

 

 

 

 

|g1r − g1r |

(4.28)

 

 

= 2

 

 

|r −, r

| dr ϕ(g1r) = Jˆgˆ ϕ(r)

 

 

 

 

 

 

ρ(r

r )

 

 

 

 

 

4.1

One-electron Approximation for Crystals

111

 

gˆKˆ ϕ(r) =

 

ρ(g1r, r )

 

 

 

 

ϕ(r )dr

 

=

|g1r − r |

(4.29)

 

|r −, r | ϕ(g1r )dr = Kˆgˆ ϕ(r)

 

 

ρ(r r )

 

 

 

 

The relation (4.27) and the equality |g1r − g1r | = |r − r | and d(g1r ) = dr

ˆ

are used to prove the invariance of F. As (4.28) and (4.29) hold for an arbitrary function ϕ(r)

 

 

ˆ

ˆ

ˆ

ˆ

 

 

(4.30)

 

 

gˆJ = Jˆg,

gˆK = Kˆg

 

and therefore

 

ˆ

ˆ

ˆ(p)

ˆ

 

(4.31)

 

 

(p)

 

 

gˆF = Fˆg,

gˆF

= F

 

ˆ ˆ

(p)

) in the electron equation (4.21) has the symmetry group G

Thus, the operator F(F

 

of the equilibrium nuclear configuration if the electron-density matrix ρ(r, r )[ρ(p)(r, r )]

ˆ ˆ

(p)

) form the bases of

is invariant under G. The eigenfunctions of the operator F(F

 

irreps of G. The invariance of ρ(r, r )[ρ(p)(r, r )] is assured for the system with closed shells.

Thus, taking as the initial approximation that the electron density ρ(0)(r, r ) is invariant under the group G we have, during any iteration step and in the self-consistent limit, the one-electron functions classified according to the irreps of the group G – the symmetry group of the equilibrium nuclear configuration.

In the absence of external fields the realness of the one-electron approximation Hamiltonian is assured at all stages of the self-consistent calculations, at least for

ˆ

(p)

leads to the real

a system with closed shells. The realness of the Hamiltonian F

 

(p+1) ˆ(p)

electron-density matrix ρ (r, r ) generated by the eigenfunctions of F . Indeed, let the functions ϕ(iβ)(r) and (ϕ(iβ)(r)) belong to the same one-electron energy and describe the occupied electron states. If they span the space of the same irrep D(β) of G (D(β) is real) then the partial sum (4.26 ) formed by these functions and involved in ρ(p+1)(r, r ) is obviously real. If they are the independent bases of the irreps D(β) and

D(β) (equivalent or inequivalent) then the sum of two terms of the type (4.26) is also real. This is the case of additional degeneracy due to the Hamiltonian being real.

Thus, the electron-density matrix is the sum of real members and is real itself. The real

(p+1) ˆ(p+1) ˆ (p+1)

density matrix ρ (r, r ) generates the real Coulomb J and exchange K

ˆ(p+1)

for the (p + 1)th

operators and therefore the real one-electron Hamiltonian F

iteration of the self-consistent calculations. The realness of the one-electron approximation Hamiltonian of molecular systems causes additional degeneracy of the energy levels in cyclic point-symmetry groups Cn. In the case of crystals degenerated energy levels appear corresponding to k and −k even for the case when the inversion is absent in the point group Fk. In the next sections we consider the Hartree–Fock LCAO approach for periodic systems in comparison with that for molecular systems.

4.1.3 Restricted and Unrestricted Hartree–Fock LCAO Methods for Molecules

The solution of Hartree–Fock equations without any additional approximations is practically possible only for atoms. Due to the high symmetry of atomic systems these equations can be solved numerically or by using analytical representation of

112 4 Hartree–Fock LCAO Method for Periodic Systems

atomic orbitals (AO). The tables of Roothaan–Hartree–Fock atomic wavefunctions by Clementi and Roetti [65] give atomic functions expanded in terms of Slater-type orbitals with integer quantum numbers. In molecular quantum chemistry the molecular orbital (MO) is expanded in terms of Gaussian-type orbitals (GTO). Di erent atomic functions used in calculations of molecules and crystals are considered in Chap. 8. For the moment, we restrict ourselves by representation of MO ϕi(r) as a linear combination of atomic orbitals χµA(r) (MO LCAO approximation):

ϕi(r) = CχµA(r)

(4.32)

µA

 

where µ numbers all basis functions centered on atom A and summation is made over all the atoms in the molecular system. The MO LCAO approximation (also known as the Hartree–Fock–Roothaan approximation [66]) is practically the only way to make first-principles calculations for molecular systems. In the standard derivation of the Hartree–Fock equations relative to a closed-shell system, the constraint that each molecular orbital is populated by two electrons or vacant is introduced (Restricted Hartree–Fock theory – RHF).

In the MO LCAO approximation RHF method (4.21) transform to the matrix equations

FC = SCE

(4.33)

where F and S are the Fock and the overlap matrices, C and E are the matrices of eigenvectors and eigenvalues. The dimension M of square matrices F, S, C is equal to the number of items in the sum (4.32), i.e. the total number of AO used in the calculation.

The Fock matrix F is the sum of one-electron H and two-electron G parts. The former includes the kinetic (T) and nuclear attraction (Z) energy, the latter is connected with the electron–electron interactions. The matrix elements Hµν are written in the form

 

 

 

 

 

|

|

 

Hµν = 1/2

χµ(r) r2 χν (r)dV +

 

χµ(r)

 

ZA

 

χν (r)dV = Tµν + Zµν

A

r

RA

 

(4.34) The one-electron matrix H is independent of the density matrix and therefore does not change during the self-consistent solution of Hartree–Fock-Rooothaan equations. The two-electron matrix G includes Coulomb (C) and exchange (X) parts of interelectron interaction:

 

 

 

1

 

 

Gµν = Cµν + Xµν = Pλσ(µν|λσ)

2

Pλσ(µλ|νσ)

(4.35)

 

 

λσ

 

λσ

 

where

 

 

 

 

 

(µν|λσ) =

χµ(r)χν (r)|r − r |1χλ(r )χσ(r )dV dV

(4.36)

are two-electron integrals calculated with basis atomic orbitals. The one-electron kinetic energy and nuclear attraction integrals and two-electron integrals can be calculated at the beginning of solution of these equations and stored in the external memory of a computer. In some cases, this integrals calculation is more e cient at each step of the self-consistent process.

4.1 One-electron Approximation for Crystals

113

The two-electron part (4.35) contains one-electron density matrix elements

 

Pλσ = 2

i

CC

(4.37)

The one-electron density matrix P in AO basis is calculated self-consistently after the initial set of coe cients C is known. The simple one-electron Hamiltonian of Huckel type is often used as an intial approximation to the Hartree–Fock Hamiltonian. It is important to remember that the one-electron approximation is made when the manyelectron wavefunction is written as the Slater determinant or their linear combination. For bound states, precise solution of Schrodinger equation can be expressed as a (in general infinite) linear combination of Slater determinants.

The approximation of the one-electron Hamiltonian is the next step in the framework of the one-electron approximation – the electron–electron interactions are excluded from the Hamiltonian. In solid-state theory the LCAO one-electron Hamiltonian approximation is known as the tight binding method. In molecular quantum chemistry the one-electron Hamiltonians of H¨uckel or Mulliken–R¨udenberg types (see Chap. 6) were popular in the 1950s and the beginning of the 1960s when the firstprinciples, Hartree–Fock LCAO calculations were practically impossible.

The total electron energy Ee in the Hartree–Fock LCAO method is calculated

from the relation

1

 

 

Ee =

Pµν (Fµν + Tµν + Zµν )

(4.38)

 

2

 

 

µν

 

The total energy E = Ee + EN is the sum of electron energy and nuclear repulsion

energy

|

 

− |

 

 

 

 

EN =

 

Zj Zj

(4.39)

 

Rj

Rj

j<j

 

 

 

 

 

 

The energy EN is fixed for fixed nuclei configuration {R1, . . . , Rj , . . . , RN }. When the optimization of molecular atomic configuration is made, the di erent nuclei configurations are considered to find that which gives the total energy minima. For each such configuration the self-consistent electron-energy calculation is made. By definition, in the molecular states with closed shells all MO are fully occupied by electrons (by 2, 4 and 6 electrons for nondegenerated, twoand three-times degenerated irreps of point groups, respectively) or empty. The corresponding many-electron wavefunction can be written as a single Slater determinant, each MO is occupied by an equal number of electrons with α and β spins. Such a function describes the ground state of a molecule with total spin S=0 and with the symmetry of identity representation of the point-symmetry group.

In the case of open-shell molecular systems a single Slater determinant describes a state with the fixed spin-projection (equal to the di erence of nα and nβ electrons) but is not the correct spin eigenfunction. Indeed, let in the open-shell configuration the highest one-electron energy level be occupied by one electron with α spin. As there are no spin interactions in the Hartree–Fock Hamiltonian the same electron energy corresponds to the function with β-spin electrons on the highest occupied level. This means that in order to get the correct total spin eigenfunction transforming over the identity representation of the point group it is necessary to use a sum of Slater

114 4 Hartree–Fock LCAO Method for Periodic Systems

determinants. The molecules with an odd number of electrons, the radicals and the magnetic systems have the open shells in the ground state.

The restricted open-shell Hartree–Fock (ROHF) and the unrestricted Hartree– Fock Method ( UHF) approximations permit, however, open-shell systems to be described, while maintaining the simplicity of the single-determinant approximation. This is made at the stage of self-consistent electronic-structure calculations. Afterwards, the obtained spin-orbitals can be used to get the correct total spin manydeterminant wavefunction and to calculate the corresponding electron energy.

The ROHF [67] many-electron wavefunction is, in the general case, a sum of Slater determinants; each determinant contains a closed-shell subset, with doubly occupied orbitals and an open-shell subset, formed by orbitals occupied by a single electron.

In one particular case, the ROHF wavefunction reduces to a single determinant: this is the so-called half-closed- shell cases, where it is possible to define two sets of orbitals, the first nd occupied by paired electrons and the second ns, by electrons with parallel spins. The total number of electrons n = nd + ns. In all molecular programs ROHF means a single-determinant wavefunction with maximal spin projection that is automatically eigenfunctions of S2 with the maximal spin projecton value S = ns/2. So, for the ROHF method projection on a pure spin state is not required. The space symmetry of the Hamiltonian in the ROHF method remains the same as in the RHF method, i.e. coincides with the space symmetry of nuclei configuration. The double-occupancy constraint allows the ROHF approach to obtain solutions that are eigenfunctions of the total spin operator. The molecular orbitals diagram for the ROHF half-closed shell is given in Fig. 4.1, (left).

DE

KnB

 

 

 

 

 

 

 

K

 

 

 

 

 

 

 

KC

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K

B

 

nC

nC+1

 

 

 

 

 

 

 

 

 

 

 

 

K

 

 

 

nB

KC

 

 

 

K

B

 

 

n

C

 

 

 

 

 

nC 1

 

 

 

nB 1

 

KnC 1

 

 

 

 

 

 

K3C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K

 

 

 

K3B

K2C

 

 

 

 

 

 

 

 

 

 

K2B

 

 

 

 

 

2

 

 

 

 

KC

K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B

 

1

1

 

 

 

K1

 

Fig. 4.1. The one-electron levels filling in (a) ROHF and (b) UHF methods

In the UHF method, keeping a single-determinant description, the constraint of double occupancy of molecular orbitals is absent as α electrons are allowed to occupy orbitals other than those occupied by the β electrons, see Fig. 4.1 (right). The