
The SelfConsistent Field (SCF) Method 

Simply, to calculate a potential energy surface, we must solve the electronic Schrödinger equation (equations (3.3)—(3.5)) for a system of n electrons and N nuclei, over a range of nuclear coordinates. This is termed an ab intio method, since it is derived from ‘first principles’. Generally, the real wavefunction of a system is too complex to be found directly, but can be approximated by a simpler wavefunction. This then enables the electronic Schrödinger equation to be solved numerically. The selfconsistent field method is an iterative method that involves selecting an approximate Hamiltonian, solving the Schrödinger equation to obtain a more accurate set of orbitals, and then solving the Schrödinger equation again with theses until the results converge. 7.1 HartreeFock theoryThe HartreeFock (HF) method [14],[15] invokes what is known as the (molecular) orbital approximation: The wavefunction is taken to be a product of oneelectron wavefunctions (equation (7.1)):
These oneelectron wavefunctions are also called orbitals. In the case of a molecule, the orbitals are expanded as atomic functions, according to a basis set:
The molecular orbital approximation assumes that the electrons behave independently of each other (equation (7.3) shows that the probability density for an electron does not depend on other electrons), and that the electronic Hamiltonian can be expressed as a sum of oneelectron Hamiltonians (equation (7.4)).
The wavefunctions are written as antisymmetrised products of spinorbitals (equations (7.5) and (7.6), where SD denotes that the wavefunction is written as a Slater determinant (see Levine[7])) in order to satisfy the Pauli principle, since the Hartree method does not ordinarily account for spin.
Pauli showed that relativistic quantum field theory indicates that particles with halfintegral spin (e.g. s=½), such as electrons, require antisymmetric wavefunctions:
In spinorbital terms, if two electrons have the same spin, the wavefunction becomes zero. E.g. If r_{1} = r_{2} in equation (7.7):
The continuous nature of (7.8) means that the probability of finding two electrons with the same spin close to each other is very small. See Levine[7] or Szabo and Ostlund[16] for a greater discussion of the Pauli principle. The HartreeFock energy is given from the energy of the Slater determinant, by equation (7.9):
In order to find the best wavefunction, the variational method is used. This relies on the variational principle, that the approximate HartreeFock wavefunction is always greater in energy than the exact ground state energy of the system (equation (7.10) – the complex conjugate of the wavefunction is multiplied by the wavefunction (integrated over all space) to normalise the probability density (recall equation (2.13))).
The energy of the HartreeFock wavefunction depends on the atomic orbital
coefficients, c_{ij}, (equation
(7.2)) used in the linear combination of atomic orbitals to construct
the molecular wavefunction (orbital).
Being an SCF method, the HartreeFock method then uses this wavefunction
again to construct another, until the energies converge. 7.2 Perturbation theoryAn alternative to the variational theorem used to minimise the energy of the Slater determinant coefficients in the HartreeFock method is perturbation theory. This is discussed in detail in Levine[7] and Szabo and Ostlund,[16] but is simply the fact that an approximation to an unsolvable Hamiltonian lies in a simpler Hamiltonian (the example given in Levine is the case of harmonic and anharmonic oscillators). The simpler Hamiltonian (superscript zero in equation (7.12), below) is called the unperturbed system, and the more accurate Hamiltonian the perturbed system. The difference between the two is the perturbation, often denoted by a prime:
7.2.1 MøllerPlesset (MP) perturbation theory A common form of socalled manybody perturbation theory is
MøllerPlesset (MP) perturbation theory,
formulated in 1934 by Møller and Plesset.[17]
In MP theory, the HF wavefunction is the unperturbed system.
…where E^{(1)} and E^{(2)}
denote first and secondorder corrections to the zerothorder (unperturbed)
wavefunctions. 7.2.2 Coupledcluster (CC) perturbation theory Another popular perturbation correction method is the coupledcluster
(CC) method, (also termed the coupledcluster
approximation (CCA)) suggested in 1958 by Coester and Kümmel.[18],[19]
Originally suggested for problems in nuclear physics, this was applied
to the quantum chemistry manybody problem by Cízek.[20] 7.3 The MultiConfiguration SelfConsistent Field (MCSCF) Method The multiconfiguration SCF (MCSCF) method
involves choosing nonSCF molecular orbitals and varying these so as to
minimise the energy. The molecular wavefunction is again written as a
linear combination of the configuration state functions, varying the coefficients
and also the forms of the molecular orbitals. 7.4 The Complete Active Space SelfConsistent Field (CASSCF) Method The most commonly used form of MCSCF calculations is the complete
active space SCF (CASSCF) method. 

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[14] D.R. Hartree, Proc. Cambridge Phil. Soc., 24, 111, 426, 1928  25, 225, 310, 1929 [15] V. Fock, Z. Physik, 61, 126, 1930 [16] Attila Szabo and Neil S. Ostlund, “Modern Quantum Chemistry”, 1st edition, rev., McGrawHill, Inc., 1989 [17] C. Møller and M.S. Plesset, Phys. Rev., 46, 618, 1934 [18] F. Coester, Nucl. Phys., 7, 421, 1958 [19] F. Coester and H. Kümmel, Nucl. Phys., 17, 477, 1960 [20] J. Cízek, J. Chem. Phys., 45, 4256, 1966 [21] Paul G. Mezey, “Potential Energy Hypersurfaces”, Studies in physical and theoretical chemistry, vol. 53, Elsevier Science Publishers B.V., 1987 [22] B.O. Roos, Adv. Chem. Phys., 69, 399, 1987 
Potential Energy Surfaces and Conical Intersections • June 2002 • Ian Grant 