The Basis Set Approximation

3 Electronic Structure

3.5 The Basis Set Approximation

For small highly symmetric systems, such as atoms and diatomic molecules, the Hartree–Fock equations may be solved by mapping the orbitals on a set of grid points, and these are referred to as numerical Hartree–Fockmethods.⁸However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis functions may in principle be used:

E_N^k₊1−E_N=e_k

EN EN hk Jik Kik k i

N i

− −¹= +

∑

( − )=^e

=1 elec

EN EN hk Jik Kik Jkj Kkj j

N i

N i i

− −¹= +¹₂

∑

( − )+¹₂

∑

( − )

=1

=1

elec elec

E h J K V

E h J K V

N i ij ij

j N i N i

N

N i ij ij

j N i N i

N

= + ( − )+

= + ( − )+

∑

∑

∑

∑

∑

∑

−

−

−

−

1 2 1

1 2

1 1 1

nn

=1

=1

=1

nn

=1

=1

=1

elec elec elec

elec elec elec

exponential, Gaussian, polynomial, cube functions, wavelets, plane waves, etc. There are two guidelines for choosing the basis functions. One is that they should have a behaviour that agrees with the physics of the problem, since this ensures that the convergence as more basis functions are added is reasonably rapid. For bound atomic and molecular systems, this means that the functions should go toward zero as the dis- tance between the nucleus and the electron becomes large. The second guidelineis a practical one: the chosen functions should make it easy to calculate all the required integrals.

The first criterion suggest the use of exponential functions located on the nuclei, since such functions are known to be exact solutions for the hydrogen atom. Unfor- tunately, exponential functions turn out to be computationally difficult. Gaussian func- tions are computationally much easier to handle, and although they are poorer at describing the electronic structure on a one-to-one basis, the computational advantages more than make up for this. For periodic systems, the infinite nature of the problem suggests the use of plane waves as basis functions, since these are the exact solutions for a free electron. We will return to the precise description of basis sets in Chapter 5, but for now simply assume that a set of M_basisbasis functions located on the nuclei has been chosen.

Each MO f is expanded in terms of the basis functions c, conventionally called atomic orbitals(MO =LCAO,Linear Combination of Atomic Orbitals), although they are generally not solutions to the atomic HF problem.

(3.49)

The Hartree–Fock equations (3.42) may be written as in eq. (3.50).

(3.50) Multiplying from the left by a specific basis function and integrating yields the Roothaan–Hall equations (for a closed shell system).⁹These are the Hartree–Fock equations in the atomic orbital basis, and all the M_basisequations may be collected in a matrix notation.

(3.51)

The Smatrix contains the overlap elements between basis functions, and the Fmatrix contains the Fock matrix elements. Each Fabelement contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over occupied MOs of coefficients multiplied with two-electron integrals involving the electron–electron repulsion. The latter is often written as a product of a densitymatrix and two-electron integrals.

FC SC F

=

=

= e F

S

ab a b

ab a b

c c c c F_{i i}

M

i i

M

c_{a a} c

a

a a

a

c e c

basis basis

∑

⁼

∑

f aca a

i i

M

=

∑

^basisc

(3.52)

For use in Section 3.8, it can also be written in a more compact notation.

(3.53) Here G⋅Ddenotes the contraction of the Dmatrix with the four-dimensional Gtensor.

The total energy (eq. (3.32)) in term of integrals over basis functions is given in eq. (3.54).

(3.54)

The latter expression may also be written as in eq. (3.55).

(3.55) The one- and two-electron integrals in the atomic basis are given as eq. (3.24).

(3.56)

The two-electron integrals are often written in a notation without electron coordinates or the goperator present.

(3.57) ca 1cg 2 1 cb cd c c c ca g b d

1

1 2

1 2

( ) ( )

 −

 

 ( ) ( ) =

∫

_{r r} ^{2 d dr r}

c c c c c c

c c c c c c c c

a b a b a b

a g b d a g b d

h r

R r r

g r r r r

= ( )

(

− ∇

)

( ) + ( )

 −

 

 ( )

= ( ) ( )

 −

 

 ( ) ( )

∫ ∑ ∫

∫

1 1 1 1

1 2 1

1

1 2

2 1

1

1

1 2

1 2

d d

2 d d

nucle Za

a a

N i

E D h D D D D V

M M

=

∑

^{a ab}+

∑

( − ) +

ab

a gd ad gb

abgd

a g b d

b 1 b c c c c

2

basis basis

g nn

E V

c c c c c c V

D

i i i

i N

i j i j i j j i

ij N

i i M i N

i j i j

M ij N

= + ( − )+

= +  −

 

+

=

∑ ∑

∑

∑ ∑ ∑

f f f f f f f f f f

c c c c c c

c c c c

a b a b

ab

a g b d

a g b d

a g d b

abgd

h g g

h g

g

elec elec

basis

elec elec basis

nn

nn

1 2

1 2

a a ab ab

a gd abgd

a g b d a g d b

bh D Db c c c c c c c c V

M M

+ ( − )+

∑

^{basis 1}₂

∑

^{basis g g nn}

F_ab h_ab G_abgdD_gd

gd

= +

= + ⋅

∑

F h G D

c c c c c c

c c c f c f c f f c

c c c c c c c c c c

c c c c

a b a b a b

a b a b a b

a b g d a g b d a g d b

gd

a b gd a g

F h J K

h g g

h g g

h g

= + −

= + ( − )

= + ( − )

= +

∑

∑

∑

∑

j j

j

j j j j

j

j M j

c c D

j occ MO

occ MO

occ MO basis .

.

.

c

c cb d c ca g c cd b gd

gd g d

( − )

=

∑

∑

g

M

j j

D c c _j

basis

occ MO.

This is known as the physicist’snotation, where the ordering of the functions is given by the electron indices. They may also be written in an alternative order with both functions depending on electron 1 on the left, and the functions depending on elec- tron 2 on the right, this is known as the Mullikenor chemist’snotation.

(3.58) The bra-ket notation has the electron indices 〈12|12〉, while the parenthesis notation has the order (11|22). In many cases the integrals are written with only the indices given, i.e.〈cacb|cgcd〉 = 〈ab|gd〉. Since Coulomb and exchange integrals often are used as their difference, the following double-bar notations are also used frequently.

(3.59) The Roothaan–Hall equation (3.51) is a determination of the eigenvalues of the Fock matrix (see Section 16.2.3 for details). To determine the unknown MO coefficients cai, the Fock matrix must be diagonalized. However, the Fock matrix is only known if all the MO coefficients are known (eq. (3.52)). The procedure therefore starts off by some guess of the coefficients, forms the Fmatrix, and diagonalizes it. The new set of coef- ficients is then used for calculating a new Fock matrix, etc. This is continued until the set of coefficients used for constructing the Fock matrix is equal to those resulting from the diagonalization (to within a certain threshold). This set of coefficients determines a self-consistent field solution.

c c c c c c c c c c c c c c c c c c c c c c c c

a b g d a b g d a b d g

a b g d a b g d a g b d

= −

( )=( )−( )

ca1cb1 1 cg cd c c c ca b g d

2

1 2

1 2

( ) ( )

 −

 

 ( ) ( ) =( )

∫

_{r r} ^{2 d dr r}

Obtain initial guess for density matrix

Form Fock matrix

Diagonalize Fock matrix

Form new density matrix

Two-electron integrals

Iterate

Figure 3.3 Illustration of the SCF procedure

The potential (or field) generated by the SCF electron density is identical to that produced by solving for the electron distribution. The Fock matrix, and therefore the total energy, only depends on the occupied MOs. Solving the Roothaan–Hall equa- tions produces a total of M_basisMOs, i.e. there are N_elecoccupied and M_basis−N_elecunoc- cupied, or virtual, MOs. The virtual orbitals are orthogonal to all the occupied orbitals,

but have no direct physical interpretation, except as electron affinities (via Koopmans’

theorem).

In order to construct the Fock matrix in eq. (3.51), integrals between all pairs of basis functions and the one-electron operator hare needed. For M_basisfunctions there are of the order of M²_basis such one-electron integrals.These one-electron integrals are also known as coreintegrals, as they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the gtwo-electron operator. There are of the order of M⁴_basisof these two-electron integrals. In conventional HF methods, the two-electron integrals are cal- culated and saved before the SCF procedure is begun, and is then used in each SCF iteration. Formally, in the large basis set limit the SCF procedure involves a computational effort that increases as the number of basis functions to the fourth power. It will be shown below that the scaling may be substantially smaller in actual calculations.

For the two-electron integrals, the four basis functions may be located on one, two, three or four different atomic centres. It has already been mentioned that exponen- tial-type basis functions (cexp(−ar)) are fundamentally better suited for electronic structure calculations. However, it turns out that the calculation of especially three- and four-centre two-electron integrals is very time-consuming for exponential func- tions. Gaussian functions (cexp(−ar²)) are much easier for calculating two-electron integrals. This is due to the fact that the product of two Gaussians located at two dif- ferent positions (R_Aand R_B) with different exponents (a and b) can be written as a single Gaussian located at an intermediate position R_C between the two original.

This allows compact formulas for all types of one- and two-electron integrals to be derived.

(3.60)

As the number of basis functions increases, the accuracy of the MOs improves. In the limit of a complete basis set (infinite number of basis functions), the results are iden- tical to those obtained by a numerical HF method, and this is known as the Hartree–Fock limit. This is notthe exact solution to the Schrödinger equation, only the best single-determinant wave function that can be obtained. In practical calculations, the HF limit is never reached, and the term Hartree–Fock is normally used also to cover SCF solutions with an incomplete basis set.Ab initioHF methods, where all the necessary integrals are calculated from a given basis set, are one-dimensional. As the

G G

G G K

K

A

B

A B

C A B

e e e

e

A

B

C

A B

r r r r

R R R

r R

r R

r R

R R

( )_{= } ^_ ( )_{= } ^_ ( ) ( )=

= +

= +

+

=  

 ( )

− ( + )

− (+ )

− (+ )

− + ( − )

2 2

2

3 4

3 4

2 3 4

2

2

2

a p

b p g a b

a b

a b p ab

a

b

g

ab a b

2 2

size of the basis set is increased, the variational principle ensures that the results become better (at least in an energetic sense). The quality of a result can therefore be assessed by running calculations with an increasingly larger basis set.