The fundamental equation of nonrelativistic quantum chemistry is the timeindependent Schrödinger equation,
(4.1) 
In quantum chemistry, this equation is solved as a function of the electronic variables (), for fixed values of the nuclear coordinates (). The Hamiltonian operator in Eq. eq400, in atomic units, is
(4.2) 
where
(4.3) 
In Eq. eq:Hamiltonian, is the nuclear charge, is the ratio of the mass of nucleus to the mass of an electron, is the distance between nuclei and , is the distance between the th and th electrons, is the distance between the th electron and the th nucleus, is the number of nuclei and is the number of electrons. The total energy is an eigenvalue of , with a corresponding eigenfunction (wave function) .
Separating the motions of the electrons from that of the nuclei, an idea originally due to Born and Oppenheimer,[Born and Oppenheimer(1927)] yields the electronic Hamiltonian operator:
(4.4) 
The solution of the corresponding electronic Schrödinger equation,
(4.5) 
affords the total electronic energy, , and electronic wave function, , which describes the distribution of the electrons for fixed nuclear positions. The total energy is obtained by simply adding the nuclear–nuclear repulsion energy [the fifth term in Eq. (4.2)] to the total electronic energy:
(4.6) 
Solving the eigenvalue problem in Eq. (4.5) yields a set of eigenfunctions (, , ) with corresponding eigenvalues .
Our interest lies in determining the lowest eigenvalue and associated eigenfunction which correspond to the ground state energy and wave function of the molecule. However, solving Eq. (4.5) for other than the most trivial systems is extremely difficult and the best we can do in practice is to find approximate solutions.
The first approximation used to solve Eq. eq404 is the independentelectron (meanfield) approximation, in which the wave function is approximated as an antisymmetrized product of oneelectron functions, namely, the MOs. Each MO is determined by considering the electron as moving within an average field of all the other electrons. This affords the wellknown Slater determinant wave function[Slater(1929), Slater(1930)]
(4.7) 
where , a spin orbital, is the product of a molecular orbital and a spin function ( or ).
One obtains the optimum set of MOs by variationally minimizing the energy in what is called a “selfconsistent field” or SCF approximation to the manyelectron problem. The archetypal SCF method is the HartreeFock (HF) approximation, but these SCF methods also include KSDFT (Chapter 5). All SCF methods lead to equations of the form
(4.8) 
where the Fock operator for the th electron is
(4.9) 
Here are spin and spatial coordinates of the th electron, the functions are spin orbitals and is the effective potential “seen” by the th electron, which depends on the spin orbitals of the other electrons. The nature of the effective potential depends on the SCF methodology, i.e., on the choice of densityfunctional approximation.
The second approximation usually introduced when solving Eq. eq404 is the introduction of an AO basis linear combinations of which will then determine the MOs. There are many standardized, atomcentered Gaussian basis sets and details of these are discussed in Chapter 8.
After eliminating the spin components in Eq. (4.8) and introducing a finite basis,
(4.10) 
Eq. (4.8) reduces to the RoothaanHall matrix equation
(4.11) 
Here, is the Fock matrix, is a square matrix of molecular orbital coefficients, is the AO overlap matrix with elements
(4.12) 
and is a diagonal matrix containing the orbital energies. Generalizing to an unrestricted formalism by introducing separate spatial orbitals for and spin in Eq. eq406 yields the PopleNesbet equations[Pople and Nesbet(1954)]
(4.13) 
In SCF methods, an initial guess is for the MOs is first determined, and from this, an average field seen by each electron can be calculated. A new set of MOs can be obtained by solving the RoothaanHall or PopleNesbet eigenvalue equations, resulting in the restricted or unrestricted finitebasis SCF approximation. This procedure is repeated until the new MOs differ negligibly from those of the previous iteration. The HartreeFock approximation for the effective potential in Eq. eq408 inherently neglects the instantaneous electronelectron correlations that are averaged out by the SCF procedure, and while the chemistry resulting from HF calculations often offers valuable qualitative insight, quantitative energetics are often poor. In principle, the DFT methodologies are able to capture all the correlation energy, i.e., the difference in energy between the HF energy and the true energy. In practice, the bestavailable density functionals perform well but not perfectly, and conventional postHF approaches to calculating the correlation energy (see Chapter 6) are often required.
That said, because SCF methods often yield acceptably accurate chemical predictions at low to moderate computational cost, selfconsistent field methods are the cornerstone of most quantumchemical programs and calculations. The formal costs of many SCF algorithms is , that is, they grow with the fourth power of system size, . This is slower than the growth of the cheapest conventional correlated methods, which scale as or worse, algorithmic advances available in QChem can reduce the SCF cost to in favorable cases, an improvement that allows SCF methods to be applied to molecules previously considered beyond the scope of ab initio quantum chemistry.
Types of groundstate energy calculations currently available in QChem are summarized in Table 4.1.
Calculation 
$rem Variable JOBTYPE 

Single point energy (default) 
SINGLE_POINT or SP 

Force (energy + gradient) 
FORCE 

Equilibrium structure search 
OPTIMIZATION or OPT 
(Ch. 10) 
Transition structure search 
TS 
(Ch. 10) 
Intrinsic reaction pathway 
RPATH 
(Sec. 10.5) 
Potential energy scan 
PES_SCAN 
(Sec. 10.4) 
Vibrational frequency calculation 
FREQUENCY or FREQ 

Polarizability and relaxed dipole 
POLARIZABILITY, DIPOLE 

NMR chemical shift 
NMR 
(Sec. 11.13.1) 
Indirect nuclear spinspin coupling 
ISSC 
(Sec. 11.13.1) 
Ab initio molecular dynamics 
AIMD 
(Sec. 10.7) 
Ab initio path integrals 
PIMD, PIMC 
(Sec. 10.8) 
BSSE (counterpoise) correction 
BSSE 
(Sec. 13.4.3) 
Energy decomposition analysis 
EDA 
(Sec. 13.5) 
As with much of the theory underlying modern quantum chemistry, the HF approximation was developed shortly after publication of the Schrödinger equation, but remained a qualitative theory until the advent of the computer. Although the HF approximation tends to yield qualitative chemical accuracy, rather than quantitative information, and is generally inferior to many of the DFT approaches available, it remains as a useful tool in the quantum chemist’s toolkit. In particular, for organic chemistry, HF predictions of molecular structure are very useful.
Consider once more the RoothaanHall equations, Eq. eq:RoothaanHall, or the PopleNesbet equations, Eq. eq:PopleNesbet, which can be traced back to Eq. eq407, in which the effective potential depends on the SCF methodology. In a restricted HF (RHF) formalism, the effective potential can be written as
(4.16) 
where the Coulomb and exchange operators are defined as
(4.17) 
and
(4.18) 
respectively. By introducing an atomic orbital basis, we obtain Fock matrix elements
(4.19) 
where the core Hamiltonian matrix elements
(4.20) 
consist of kinetic energy elements
(4.21) 
and nuclear attraction elements
(4.22) 
The Coulomb and exchange elements are given by
(4.23) 
and
(4.24) 
respectively, where the density matrix elements are
(4.25) 
and the two electron integrals are
(4.26) 
Note: The formation and utilization of twoelectron integrals is a topic central to the overall performance of SCF methodologies. The performance of the SCF methods in new quantum chemistry software programs can be quickly estimated simply by considering the quality of their atomic orbital integrals packages. See Appendix B for details of QChem’s AOInts package.
Substituting the matrix element in Eq. (4.19) back into the RoothaanHall equations, Eq. eq:RoothaanHall, and iterating until selfconsistency is achieved will yield the RHF energy and wave function. Alternatively, one could have adopted the unrestricted form of the wave function by defining separate and density matrices:
(4.27) 
The total electron density matrix . The unrestricted Fock matrix,
(4.30) 
differs from the restricted one only in the exchange contributions, where the exchange matrix elements are given by
(4.31) 