where vs (r) is an as yet undetermined potential. The solution can be ob-

tained by the variational principle, e.g., by expanding the functions φi in a

suitable set of basis functions. Slater-type Gaussian basis sets may be used,

but it is also possible to use a basis set of simple plane waves, particularly

if the system under study is periodic.

104 Quantum chemistry: solving the time-independent Schr¨dinger equation

o

In order to ¬nd expressions for vs (r), we ¬rst note that the energy func-

tional of the non-interacting system is given by

E[ρ] = Ks [ρ] + vs (r)ρ(r) dr. (4.66)

The energy functional of the real interacting system is given by (4.59). Now

writing the potential vs (r) in the hamiltonian for the Kohn-Sham orbitals

(4.65) as

ρ(r )

vs (r) = vn (r) + dr + vxc (r), (4.67)

|r ’ r |

the Kohn“Sham wave functions (or their expansion coe¬cients in the chosen

basis set) can be solved. In this potential the nuclear potential and the

electrostatic electronic interactions are included; all other terms (due to

electron correlation, exchange and the di¬erence between the real kinetic

energy and the kinetic energy of the non-interacting electrons) are absorbed

in the exchange-correlation potential vxc . The equations must be solved

iteratively until self-consistency because they contain the charge density

that depends on the solution. Thus the Kohn“Sham equations are very

similar to the SCF equations of Hartree“Fock theory.

As long as no theory is available to derive the form of the exchange-

correlation potential from ¬rst principles, approximations must be made.

In its present implementation it is assumed that vxc depends only on local

properties of the density, so that it will be expressible as a function of the

local density and its lower derivatives. This excludes the London dispersion

interaction, which is a correlation e¬ect due to dipole moments induced by

distant ¬‚uctuating dipole moments. The ¬rst attempts to ¬nd a form for the

exchange-correlation functional (or potential) started from the exact result

for a uniform electron gas, in which case the exchange potential is inversely

proportional to the cubic root of the local density:

1/3

3 3

=’

LDA

ρ1/3

vx (4.68)

4 π

so that the exchange functional Ex equals

1/3

3 3

=’

LDA

ρ4/3 (r) dr.

Ex [ρ] (4.69)

4 π

This local density approximation (LDA) is not accurate enough for atoms

and molecules. More sophisticated corrections include at least the gradient

of the density, as the popular exchange functional proposed by Becke (1988,

1992). With the addition of a proper correlation functional, as the Lee,

4.8 Excited-state quantum mechanics 105

Yang and Parr functional (Lee et al., 1988), which includes both ¬rst and

second derivatives of the density, excellent accuracy can be obtained for

structures and energies of molecules. The combination of these exchange and

correlation functionals is named the BLYP exchange-correlation functional.

A further modi¬cation B3LYP exists (Becke, 1993). The functional forms

can be found in Leach (2001).

4.8 Excited-state quantum mechanics

Normally, quantum-chemical methods produce energies and wave functions

(or electron densities) for the electronic ground state. In many applications

excited-state properties are required. For the prediction of spectroscopic

properties one wishes to obtain energies of selected excited states and tran-

sition moments between the ground state and selected excited states. For

the purpose of simulation of systems in which excited states occur, as in

predicting the fate of optically excited molecules, one wishes to describe the

potential energy surface of selected excited states, i.e., the electronic energy

as a function of the nuclear coordinates. While dynamic processes involving

electronically excited states often violate the Born“Oppenheimer approxi-

mation and require quantum-dynamical methods, the latter will make use

of the potential surfaces of both ground and excited states, generated under

the assumption of stationarity of the external potential (nuclear positions).

Within the class of Hartree“Fock methods, certain excited states, de¬ned

by the con¬guration of occupied molecular orbitals, can be selected and opti-

mized. In the con¬guration interaction (CI) scheme to incorporate electron

correlation, such excited states are considered, and used to mix with the

ground state. The popular complete active space SCF (CASSCF) method

of Roos (1980) can also be applied to speci¬c excited-state con¬gurations

and produce excited-state potential surfaces.

Unfortunately, density-functional methods are only valid for the ground

state and cannot be extended to include excited states. However, not all is

lost, as time-dependent DFT allows the prediction of excited-state proper-

ties. The linear response of a system to a periodic perturbation (e.g., an

electric ¬eld) can be computed by DFT; excited states show up by a peak

in absorbance, so that at least their relative energies and transition mo-

ments can be computed. If this is done for many nuclear con¬gurations, the

excited-state energy surface can be probed. This application is not straight-

forward, and, thus far, DFT has not been used much for the purpose of

generating excited-state energy surfaces.

106 Quantum chemistry: solving the time-independent Schr¨dinger equation

o

4.9 Approximate quantum methods

While DFT scales more favorably with system size than extended Hartree-

Fock methods, both approaches are limited to relatively small system sizes.

This is particularly true if the electronic calculation must be repeated for

many nuclear con¬gurations, such as in molecular dynamics applications.

In order to speed up the electronic calculation, many approximations have

been proposed and implemented in widespread programs. Approximations

to HF methods involve:

(i) restricting the quantum treatment to valence electrons;

(ii) restricting the shape of the atomic orbitals, generally to Slater-type

orbitals (STO) of the form rn’1 exp(’±r)Ylm (θ, φ);

(iii) neglecting or simplifying the overlap between neighboring atomic or-

bitals;

(iv) neglecting small integrals that occur in the evaluation of the Hamil-

tonian needed to minimize the expectation of the energy (4.58);

(v) replacing other such integrals by parameters.

Such methods require parametrization based on experimental (structural,

thermodynamic and spectroscopic) data and are therefore classi¬ed as semi-

empirical methods. This is not the place to elaborate on these methods; for

a review the reader is referred to Chapter 5 of Cramer (2004). Su¬ce to say

that of the numerous di¬erent approximations, the MNDO (modi¬ed neglect

of di¬erential overlap) and NDDO (neglect of diatomic di¬erential overlap)

methods seem to have survived. Examples of popular approaches are AM1

(Dewar et al., 1985: the Austin Model 1) and the better parameterized PM3

(Stewart, 1989a, 1989b: Parameterized Model 3), which are among others

available in Stewart™s public domain program MOPAC 7.

Even semi-empirical methods do not scale linearly with the number of

atoms in the system, and are not feasible for systems containing thousands

of atoms. For such systems one looks for linear-scaling methods, such as

the DAC (“divide-and-conquer”) DFT scheme of Yang (1991a, 1991b). In

this scheme the system is partitioned into local areas (groups of atoms, or

even atoms themselves), where the local density is computed directly from

a density functional, without evoking Kohn“Sham orbitals. One needs a

local Hamiltonian which is a projection of the Hamiltonian onto the local

partition. The local electron occupation is governed by a global Fermi level

(electron free energy), which is determined by the total number of electrons

in the system. This description has been improved by a formulation in terms

of local density matrices (Yang and Lee, 1995) and promises to be applicable

to very large molecules (Lee et al., 1996).

4.10 Nuclear quantum states 107

For solids, an empirical approach to consider the wave function as a linear

combination of atomic orbitals with ¬tted parameters for the interactions

and overlap, is known under the name tight-binding approximation. The TB

approximation is suitable to be combined with molecular dynamics (Laaso-

nen and Nieminen, 1990).