3 Electronic Structure
3.10 Semi-Empirical Methods
The cost of performing an HF calculation scales formally as the fourth power of the number of basis functions. This arises from the number of two-electron integrals nec- essary for constructing the Fock matrix. Semi-empirical methods reduce the computa- tional cost by reducing the number of these integrals.41 Although linear scaling methods can reduce the scaling of ab initioHF methods to ~Mbasis, this is only the lim- iting behaviour in the large basis set limit, and ab initiomethods will still require a sig- nificantly larger computational effort than semi-empirical methods.
The first step in reducing the computational problem is to consider only the valence electrons explicitly; the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third
rows of the periodic table have four basis functions (one s- and one set of p-orbitals, px, pyand pz). The large majority of semi-empirical methods to date use only s- and p- functions, and the basis functions are taken to be Slater type orbitals (see Chapter 5), i.e. exponential functions.
The central assumption of semi-empirical methods is the Zero Differential Overlap (ZDO) approximation, which neglects all products of basis functions that depend on the same electron coordinates when located on different atoms. Denoting an atomic orbital on centre A as mA(it is customary to denote basis functions with m,n,land s in semi-empirical theory, while we are using ca,cb,cgand cdfor ab initiomethods), the ZDO approximation corresponds to mAnB=0. Note that it is the productof functions on different atoms that is set equal to zero, not the integralover such a product. This has the following consequences (eqs (3.51 and (3.56)):
(1) The overlap matrix Sis reduced to a unit matrix.
(2) One-electron integrals involving three centres (two from the basis functions and one from the operator) are set to zero.
(3) All three- and four-centre two-electron integrals, which are by far the most numer- ous of the two-electron integrals, are neglected.
To compensate for these approximations, the remaining integrals are made into param- eters, and their values are assigned based on calculations or experimental data. Exactly how many integrals are neglected, and how the parameterization is done, defines the various semi-empirical methods.
Rewriting eq. (3.52) with semi-empirical labels gives the following expression for a Fock matrix element, where a two-electron integral is abbreviated as 〈mn|ls〉 (eq. (3.57)).
(3.80)
Approximations are made for the one- and two-electron parts as follows.
3.10.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) In the Neglect of Diatomic Differential Overlap(NDDO) approximation there are no further approximations than those mentioned above. Using mandnto denote either an s- or p-type (px, py or pz) orbital, the NDDO approximation is defined by eqs (3.81)–(3.83).
Overlap integrals (eq. (3.51)):
(3.81) One-electron operator (eq. (3.24)):
(3.82) Here Z′a denotes that the nuclear charge has been reduced by the number of core electrons.
h= − ∇ − R ′r V
− = − ∇ −
∑ ∑
1 2
2 1
2
Za 2 a a N
a a
nuclei Nnuclei
Smn= m n =d dmn AB
F h D
h
M
mn mn ls
ls mn
mn ls ml ns m n
= + ( − )
=
∑
basish
One-electron integrals (eq. (3.56)):
(3.83)
Due to the orthogonality of the atomic orbitals, the first one-centre matrix element in eq. (3.83) is zero unless the two functions are identical.
(3.84) Two-electron integrals (eq. (3.57)):
(3.85) 3.10.2 Intermediate Neglect of Differential Overlap Approximation (INDO) The Intermediate Neglect of Differential Overlap(INDO) approximation neglects all two-centre two-electron integrals that are not of the Coulomb type, in addition to those neglected by the NDDO approximations. Furthermore, in order to preserve rotational invariance, i.e. the total energy should be independent of a rotation of the coordinate system, integrals such as 〈mA|Va|mA〉and 〈mAnB|mAnB〉must be made independent of the orbital type (i.e. an integral involving a p-orbital must be the same as with an s-orbital).
This has as a consequence that one-electron integrals involving two different functions on the same atom and a Vaoperator from another atom disappear. The INDO method involves the following additional approximations, beside those for NDDO.
One-electron integrals (eq. (3.83)):
(3.86)
Two-electron integrals are approximated as in eq. (3.87), except that one-centre inte- grals 〈mAlA|nAsA〉are preserved.
(3.87) The surviving integrals are commonly denoted by g.
(3.88) The INDO method is intermediate between the NDDO and CNDO methods in terms of approximations.
3.10.3 Complete Neglect of Differential Overlap Approximation (CNDO) In the Complete Neglect of Differential Overlap (CNDO) approximation all the Coulomb two-electron integrals are subjected to the condition in eq. (3.87), including
m n m n m m m m g m n m n g
A A A A A A A A AA
A B A B AB
= =
=
m nA B l sC D =dACdBDd dml ns m nA B m nA B
m m m m m m
m n dmn m m
A A A A A A A
A
A A A A
A
nuclei
nuclei
h V V
h V
= − ∇ − −
= −
≠
≠
∑
∑
1 2
2 a
a N
a a
N
m nA B l sC D =dACdBD m nA B l sA B
mA− ∇ −12 2 VAnA =dmn mA− ∇ −12 2 VAmA
m n m n m n
m n m n
m n
A A A A A A A
A
A B A A B B
A C B
nuclei
h V V
h V V
V
= − ∇ − −
= − ∇ − −
=
∑
≠ 12 2
1 2
2
0
a a
N
the one-centre integrals, and are again parameterized as in eq. (3.88). The approxi- mations for the one-electron integrals in CNDO are the same as for INDO. The Pariser–Pople–Parr (PPP) method can be considered as a CNDO approximation where only π-electrons are treated.
The main difference between CNDO, INDO and NDDO is in the treatment of the two-electron integrals. While CNDO and INDO reduce these to just two parameters (gAA and gAB), all the one- and two-centre integrals are retained in the NDDO approximation. Within an sp-basis, however, there are only 27 different types of one- and two-centre integrals, while the number rises to over 500 for a basis contain- ing both s-,p- and d-functions.