3.2 Methods of Quantum Chemistry
3.2.1 Hartree-Fock Theory
Hartree-Fock and most other theories of electronic structure are based on the Born-Oppenheimer approximation and the neglect of relativistic effects such as potential retardations. Within these approximations, nuclei and electrons have a negligible momentum exchange because of their huge differences in mass.
Carrying similar amounts of momentum, electrons move very fast within their orbitals, compared with the motion of the nuclei. For electronic structure con-siderations it can thus be assumed that the nuclei are virtually at rest while electronic orbitals have ample time to relax into their ground states or follow any nucleic motion adiabatically.
This allows simplifications to the Hamiltonian for the electronic problem since the nuclear motion and the electrostatic nuclear repulsion can be sepa-rated. The separated nuclear electrostatic repulsion provides a constant offset to the Hamiltonian for the electronic problem and its eigenvalues. The Hamilto-nian for nuclear motion (including electron shells) then contains only the nuclear kinetic energy part, nuclear-nuclear electrostatic repulsion and an electronic po-tential, depending on nuclear separation, which is in fact the PES as resulting from the solution of the electronic problem. The nuclei, or the molecules in our (O2)2 problem, thus move within the PES, explaining its importance for all scattering effects.
The separated Hamiltonian for the electronic problem is the following (in
3.2. METHODS OF QUANTUM CHEMISTRY 51 atomic units as described in appendix C):
H=X
This Hamiltonian describes the electron kinetic energy (first term) and the Coulomb interaction of N electrons of unit negative charge and ions of charge Za. Distances are relative distances of the type rij = |ri −rj|. The second term describes the Coulomb attraction between electrons and ions, VeN, and the third term,Vee, describes the repulsion between the electrons.
The full electronic wavefunction Ψ of an atom can be approximatively de-scribed by a combination of individual single electron orbitals. This approach is known as “Linear Combination of Atomic Orbitals” (LCAO). Molecular orbitals (MO) are also formed by LCAO. Because of the fermionic nature of electrons, two electrons of equal spin cannot occupy a single orbital. Electronic spatial orbitalsψ(r) thus must have a spin component, which is usually symbolised by the two orthonormal spin (↑/↓) functionsα(ω) andβ(ω), and are then referred to as spin-orbitalsχ(x) =ψ(r)α(ω) orχ(x) =ψ(r)β(ω), wherexhas three spa-tial and one spin coordinate. In order to satisfy the fermionic Pauli exclusion principle, LCAO requires the product of individual single electron spin-orbitals to be antisymmetrised, so that an exchange of any two electrons causes a sign change in the product and so that two electrons with parallel spin are explicitly correlated.
Ψ(x1,x2,· · ·) =−Ψ(x2,x1,· · ·) (3.3) For two spin-orbitalsχ1(x1) and χ2(x2), the product state thus becomes
Ψ(x1,x2) = 1
√2[χ1(x1)χ2(x2)−χ1(x2)χ2(x1)] (3.4) For products of two or more single electronic spin-orbitals the antisymmetric product can conveniently be written as a Slater determinant, which has all the properties required by the antisymmetrisation. In particular, the determinant vanishes when any two spin-orbitals are the same. It can also be written using the ket notation as a short-hand.
Ψ(x1,x2,· · ·,xN) = (N!)−12 The combination of spin-orbitals that best describes the ground state Ψ0 will have the lowest energy eigenvalueE0 =hΨ0|H|Ψ0i in the Schr¨odinger equation HΨ =EΨ with the simplified Hartree-Fock Hamiltonian as discussed above.
By choosing an orthonormal set of spin-orbitals hχi|χji=δij, we can now minimise the energy with respect to the spin-orbitalsχi, where the i are the
orbital energies of the spin-orbitalsχi, acting as Lagrange multipliers introduced due to the minimisation side condition of normalised orbitals in the rigorous derivation (see [77]) of the Hartree-Fock equation.
∂
The resulting equations for the best spin-orbitals (minimising the energy) are the Hartree-Fock integro-differential equations sum over A in the second term calculates the Coulomb potential by A ions, static within the Born-Oppenheimer approximation. The third term is the Hartree term describing the electrostatic effect of the other electrons averaged over their individual probability distributions. The unphysical self-interaction fori=jin the Hartree term is cancelled in the fourth term, the exchange term, mandated by the Pauli principle. This term will vanish unless the two electrons have the same spin σ.
In the literature, a Fock operator f =h+X
j
[Jj− Kj] (3.8)
is often defined so that equation (3.7) can be written as f|χii =i|χii, where h is the single particle Hamiltonian for an electron moving in the field of theA ions (x-dependence omitted). J represents a Coulomb operator, averaging the interactionr12−1 of a second electron density distribution over all space and spin coordinates. The antisymmetric exchange operator K is a nonlocal operator because it does not define a simple spatial potential; it takes care of the spin exclusion requirement.
For numerical computations, the spatial part of the spin-orbitals is expanded in a finite set of N functions φwith coefficientsCki
ψi(r) =
N
X
k=1
Ckiφk(r). (3.9)
The choice of this finite set will ultimately restrict the accuracy of the Hartree-Fock molecular orbitals to the space spanned by this basis. While it seems prudent to choose the basis functions close to actual orbital functions, such as Slater orbitals proportional toe−ξ|r|, consideration of computational tractability dictate basis sets which are easy to integrate analytically, such as Gaussian orbital functions proportional to e−ζ|r|2. This is because the above expansion
3.2. METHODS OF QUANTUM CHEMISTRY 53 reduces the problem of calculating the electronic orbitals to the problem of finding the appropriate coefficients Cki for the chosen basis, which requires a large number of integrations of the basis set as we will outline below.
In a closed-shell approach, the spin-orbitals are restricted to doubly oc-cupied (opposite spin) spatial orbitals, and the spin component can be elim-inated from the Hartree-Fock equation. Using the above basis expansion in this Hartree-Fock equation and defining basis function overlap integral matrix Skl=R
whereiis the index of the basis set expansion chosen for the problem.
The set of matrix equations for the coefficients Cli can be solved numer-ically using techniques such as iterative diagonalisation, in what is called a self-consistent field (SCF) method. Starting from an initial guess at the correct spin-orbitals, iterations are carried out, consecutively replacing and improving the initial guess with the result of the previous step. This proceeds until the resulting orbitals, as defined by the set of coefficients, do not change (much) any more and the solution has thus become self-consistent in the iteration process.
It must be noted that the concept of universally labelling molecular and atomic orbitals by their angular momenta is an artifact which can be attributed to the Hartree-Fock single electron orbital approach, where the distribution of many electrons is given simply by the sum of the single electron densities. Fur-thermore the single-determinant Hartree-Fock method neglects electron corre-lations, using the simple non-local average potential of the other electrons.
These limits of Hartree-Fock theory can be improved by a number of other elaborate methods, such as for example perturbative methods, as described in 3.2.3, or using the Configuration Interaction (CI) and Coupled Cluster (CC) methods. Of the large number of methods used in quantum chemistry we will outline only the ones relevant for our work.