The Energy of a Slater Determinant - 3 Electronic Structure

3 Electronic Structure

3.3 The Energy of a Slater Determinant

In order to derive the HF equations, we need an expression for the energy of a single Slater determinant. For this purpose, it is convenient to write it as an antisymmetriz- ing operator A working on the “diagonal” of the determinant, where A can be expanded as a sum of permutations. We will denote the diagonal product by Π, and use the symbol Φto represent the determinant wave function.

ΦSD=

( ) ( ) ( )

1 =

1 1 1

2 2 2

1 2

N N N

N N

i j ij

! ;

f f f

f f d L

M M O M

L E_e= Ψ ^eΨ

Ψ Ψ H a a b b a b b a

= =

1 0

(3.21)

The 1operator is the identity, while the sum over P_ijgenerates all possible permutations of two electron coordinates, the sum over P_ijk all possible permutations of three electron coordinates, etc. It may be shown that Acommutes with H, and that A acting twice gives the same as A acting once, multiplied with the square root of N factorial.

(3.22) Consider now the Hamiltonian operator. The nuclear–nuclear repulsion does not depend on electron coordinates and is a constant for a given nuclear geometry. The nuclear–electron attraction is a sum of terms, each depending only on one electron coordinate. The same holds for the electron kinetic energy. The electron–electron repulsion, however, depends on two electron coordinates.

(3.23)

We note that the zero point of the energy corresponds to the particles being at rest (Te=0) and infinitely removed from each other (Vne=Vee=Vnn=0).

The operators may be collected according to the number of electron indices.

(3.24)

The one-electron operator h_idescribes the motion of electron iin the field of all the nuclei, and g_ijis a two-electron operator giving the electron–electron repulsion.

The energy may be written in terms of the permutation operator as (using eqs (3.21) and (3.22))

h R r

g r r

H h g V

i i a

a i

a N

i j

i i N

ij j i N

= − ∇ − Z

−

= −

= + +

∑

∑ ∑

1 2

nuclei

elec elec

e nn

H T V V V

V R r

V r r

V R R

e e ne ee nn

nn elec

elec nuclei

elec elec

nuclei nuclei

= + + +

= − ∇

= − −

= −

∑

1 2

i i N

a i

i N a N

i j

j i N i N

a b

b a N a N

Z Z AH HA

AA A

= N!

Φ= [ ( ) ( ) ( )]= Π

= ( )− =  − + −





∑



∑

−

A A

A P 1 P P

f1 f2 f

0 1

1 2

1 1 1

. . .

! ! . . .

N p

ij ijk

ijk ij p

N N

(3.25)

The nuclear repulsion operator is independent of electron coordinates and can imme- diately be integrated to yield a constant.

(3.26) For the one-electron operator only the identity operator can give a non-zero contri- bution. For coordinate 1 this yields a matrix element over orbital 1.

(3.27)

This follows since all the MOs fi are normalized. All matrix elements involving a permutation operator gives zero. Consider for example the permutation of electrons 1 and 2.

(3.28) This is zero as the integral over electron 2 is an overlap of two different MOs, which are orthogonal (eq. (3.20)).

For the two-electron operator, only the identity and P_ij operators can give non- zero contributions. A three-electron permutation will again give a least one overlap integral between two different MOs, which will be zero. The term arising from the identity operator is given by eq. (3.29).

(3.29)

The J12matrix element is called a Coulombintegral. It represents the classical repulsion between two charge distributions described by f12(1) and f²2(2). The term arising from the Pijoperator is given in eq. (3.30).

(3.30)

The K₁₂matrix element is called an exchange integral, and has no classical analogy.

Note that the order of the MOs in the Jand Kmatrix elements is according to the electron indices. The energy can thus be written as in eq. (3.31).

Πg P Π g

g g

12 12 1 2 12 2 1

1 2 12 2 1

1 2 12 2 1 12

1 2 1 2

= ( ) ( ) ( ) ( ) ( ) ( )

= ( ) ( ) ( ) ( ) =

f f f f f f

f f f f

. . . .

. . .

N N

Πg Π g

g g

12 1 2 12 1 2

1 2 12 1 2

1 2 12 1 2 12

1 2 1 2

= ( ) ( ) ( ) ( ) ( ) ( )

= ( ) ( ) ( ) ( ) =

f f f f f f

f f f f

. . . .

. . .

N N

Πh P Π h

1 12 1 2 1 2 1

1 1 2 2 1

1 2 1 2

1 1 2 2

= ( ) ( ) ( ) ( ) ( ) ( )

f f f f f f

. . . .

. . .

N N

Π Πh h

h h

1 1 2 1 1 2

1 1 1 2 2

1 1 1 1

1 2 1 2

1 1 2 2

1 1

= ( ) ( ) ( ) ( ) ( ) ( )

= ( ) ( ) =

f f f f f f

f f

. . . .

. . .

N N

ΦVnnΦ =Vnn Φ Φ =Vnn

E N

p p

∑

( )− Φ Φ

Π Π

H A H A

H A H P

! 1

(3.31)

The minus sign for the exchange term comes from the factor of (−1)^pin the antisym- metrizing operator, eq. (3.21). The energy may also be written in a more symmetrical form as in eq. (3.32).

(3.32) The factor of ¹/2allows the double sum to run over all electrons (it is easily seen from eqs (3.29) and (3.30) that the Coulomb “self-interaction”J_iiis exactly cancelled by the corresponding “exchange” element K_ii).

For the purpose of deriving the variation of the energy, it is convenient to express the energy in terms of Coulomb (J) and exchange (K) operators.

(3.33)

Note that the Joperator involves “multiplication” with a matrix element with the same orbital on both sides, while the Koperator “exchanges” the two functions on the right- hand side of the g₁₂operator.

The objective is now to determine a set of MOs that makes the energy a minimum, or at least stationary with respect to a change in the orbitals. The variation, however, must be carried out in such a way that the MOs remain orthogonal and normalized.

This is a constrained optimization, and can be handled by means of Lagrange multi- pliers(see Section 12.5). The condition is that a small change in the orbitals should not change the Lagrange function, i.e. the Lagrange function is stationary with respect to an orbital variation.

(3.34)

The variation of the energy is given by eq. (3.35).

(3.35)

δ δ δ

δ δ

E i i i i i i i

i j j i i j j i

j i i j j i i j

ij N

= ( + )+

− + − +

− + −

 



∑

f f f f

h h

J K J K

elec

1 elec

2 L E L E

ij ij N

i j ij

ij ij N

i j i j

= − ( − )

= − ( − )=

∑

l f f

l f f f f

elec

δ δ δ δ 0

E _i _i _i V

i N

j i j j i j

ij N

i j i i j

i j i j i

= + ( − )+

( ) = ( ) ( ) ( ) ( ) = ( ) ( ) ( )

∑

^f ^f

∑

^f ^f ^f ^f

f f f f

h J K

J g

K g

elec elec

1 2

2 1 1 2

12 12

E hi J K V

i N

ij ij

j N i

= + N ( − )+

= = =

∑ ∑ ∑

1 1 1

1 2

elec elec elec

E h J K V

i i

i N

i j i j i j j i

j i N i N

i i N

ij ij

j i N i N

= ( ) ( ) +

( ) ( ) ( ) ( ) − ( ) ( ) ( ) ( )

( )+

= + ( − )+

= = >

∑

∑ ∑ ∑

f f

f f f f f f f f

1 1

1 2 1 2 1 2 1 2

1 1

12 12

1 1

g g

elec

elec elec

elec elec elec

The third and fifth terms are identical (since the summation is over all iand j), as are the fourth and sixth terms. They may be collected to cancel the factor of ¹/2, and the variation can be written in terms of a Fock operator,F_i.

(3.36)

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron and the attraction to all the nuclei (h_i), as well as the repulsion to all the other electrons (via the Jand Koperators). Note that the Fock operator is associated with the variationof the total energy, not the energy itself. The Hamilton- ian operator (3.23) is nota sum of Fock operators.

The variation of the Lagrange function (eq. (3.34)) now becomes eq. (3.37).

(3.37) The variational principle states that the desired orbitals are those that make δL=0.

Making use of the complex conjugate properties in eq. (3.38) gives eq. (3.39).

(3.38)

(3.39) The variation of either〈δf|or〈δf|* should make δL=0, i.e. the first two terms in eq.

(3.39) must cancel, and the last two terms must cancel. Taking the complex conjugate of the last two terms and subtracting them from the first two gives eq. (3.40).

(3.40) This means that the Lagrange multipliers are elements of a Hermitian matrix (lij=lji*). The final set of Hartree–Fock equationsmay be written as in eq. (3.41).

(3.41) The equations may be simplified by choosing a unitary transformation (Section 16.2) that makes the matrix of Lagrange multipliers diagonal, i.e.lij =0 and lii= ei). This special set of molecular orbitals (f′) is called canonicalMOs, and transforms eq. (3.41) into a set of pseudo-eigenvalue equations.

(3.42) F_if_i′ =e f_i _i′

Fi i ij j j N

f =

∑

^elecl f lij lji f f

ij N

i j

(

−

)

⁼

∑

^elec ^* ^δ ⁰

δL δ i i i δ δ δ

i N

ij ij N

i j i i i

i N

ij ij N

j i

∑

^elec ^f ^F^f −

∑

^elec^l ^{f f} +

∑

^elec ^f ^F ^f ^*−

∑

^elec^l ^{f f} ^*=⁰ f f f f

f f f f

δ δ

F F

δL δ i i i i iδ i δ δ

i N

ij ij N

i j i j

∑

^elec( ^f ^F^f + ^f ^F ^f )−

∑

^elec^l ( ^{f f} + ^f ^f )

δ δ δ δ δ

δ δ δ

E E

i i i i i i

i N

i j j i i j j i

ij N

i i i i i i

i N

i i j j

j N

= ( + )+ ( − + − )

= ( + )

= + ( − )

∑ ∑

∑

f f f f f f f f

f f f f

h h J K J K

F F

F h J K

elec elec

elec

The Lagrange multipliers are seen to have the physical interpretation of MO energies, i.e. they are the expectation value of the Fock operator in the MO basis (multiply eq.

(3.42) by f′i* from the left and integrate).

(3.43) The Hartree–Fock equations form a set of pseudo-eigenvalue equations as the Fock operator depends on all the occupied MOs (via the Coulomb and exchange operators, eqs (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for solving the problem. A set of functions that is a solution to eq. (3.42) is called self- consistent field(SCF) orbitals.

The canonical MOs may be considered as a convenient set of orbitals for carrying out the variational calculation. The total energy, however, depends only on the total wave function, which is a Slater determinant written in terms of the occupied MOs, eq.

(3.20). The total wave function is unchanged by a unitary transformation of the occupied MOs among themselves (rows and columns in a determinant can be added and subtracted without affecting the determinant itself). After having determined the canonical MOs, other sets of MOs may be generated by forming linear combinations, such as localized MOs, or MOs displaying hybridization, which is discussed in more detail in Section 9.4.

The orbital energies can be considered as matrix elements of the Fock operator with the MOs (dropping the prime notation and letting fbe the canonical orbitals). The total energy can be written either as eq. (3.32) or in terms of MO energies (using the definition of Fin eqs (3.36) and (3.43)).

(3.44)

The total energy is notsimply a sum of MO orbital energies. The Fock operator con- tains terms describing the repulsion to all other electrons (Jand Koperators), and the sum over MO energies therefore counts the electron–electron repulsion twice, which must be corrected for. It is also clear that the total energy cannot be exact, as it describes the repulsion between an electron and all the other electrons, assuming that their spatial distribution is described by a set of orbitals. The electron–electron repulsion is only accounted for in an average fashion, and the HF method is therefore also referred to as a mean-fieldapproximation. As mentioned previously, this is due to the approximation of a single Slater determinant as the trial wave function.

Dalam dokumen Introduction to Computational Chemistry (Halaman 108-113)