Ab Initio Methods and Hartree-Fock Theory

In quantum chemistry, ab initio methods refer to the group of methods derived from the Schrödinger equation (Eq. A1).1

𝐻𝐻�Ψ = 𝐸𝐸Ψ (A1)

In the time-independent Schrödinger equation (Eq. A1) Ĥ is the Hamiltonian operator,

Ψ is the wave function and E is the total energy of the system. In quantum chemistry, the Hamiltonian operator is usually defined as the molecular or Coulomb Hamiltonian which represents the energy of the electrons and nuclei in a molecule and consists of five terms: 𝐻𝐻� = − ∑ _2𝑚𝑚ℏ2 𝑒𝑒∇𝑖𝑖 2 𝑖𝑖 − ∑ ℏ 2 2𝑚𝑚𝑘𝑘∇𝑘𝑘 2 𝑘𝑘 − ∑ ∑ 𝑒𝑒 2_𝑍𝑍 𝑘𝑘 𝑟𝑟𝑖𝑖𝑘𝑘 𝑘𝑘 𝑖𝑖 + ∑ 𝑒𝑒 2 𝑟𝑟𝑖𝑖𝑖𝑖 𝑖𝑖<𝑗𝑗 + ∑ 𝑒𝑒 2_𝑍𝑍 𝑘𝑘𝑍𝑍𝑙𝑙 𝑟𝑟𝑘𝑘𝑙𝑙 𝑘𝑘<𝑙𝑙 (A2) In the Hamiltonian operator (Eq. A2), i and j define electrons and k and l define nuclei which split the operator into the kinetic energies of the electrons and nuclei, the

8. Appendix – Theoretical Methods

Coulombic attraction between the electron and nuclei, and the Coulombic interelectronic and internuclear repulsion terms. In addition, ħ is Planck’s constant

divided by 2π, me is the mass of an electron, mkis the mass of nucleus k, e is the charge

of an electron, Z is an atomic number, rxyis the distance between particles x and y, and

finally ∇2 is the Laplacian operator which for Cartesian coordinates x, y, and z is:

∇2₌ 𝜕𝜕2 𝜕𝜕𝜕𝜕2+ 𝜕𝜕2 𝜕𝜕𝜕𝜕2+ 𝜕𝜕2 𝜕𝜕𝜕𝜕2 (A3)

Even this simplified form of the time-independent Schrödinger equation cannot be solved analytically for most systems with the exception of one-electron systems such as hydrogen atoms. Therefore further approximations must be made to simplify the calculations for larger systems.

One of the most important approximations is the separation of the nuclear and electronic terms by the Born-Oppenheimer approximation.2 This approximation is based on the significant difference in the masses between electrons and nuclei where the heavier nuclei are considered to move much slower than the electrons and thus the electrons can be assumed to be moving in a field of fixed nuclei. This simplifies Eq. A2 to the electronic Hamiltonian which does not contain the nuclear kinetic energy term and the nuclear potential energy remains constant for a particular nuclei arrangement.

Using the Born-Oppenheimer approximation, the total wave function can now instead be expressed as:

Ψ𝑡𝑡𝑡𝑡𝑡𝑡(𝑟𝑟𝑒𝑒𝑙𝑙, 𝑟𝑟𝑁𝑁) = Ψ𝑒𝑒𝑙𝑙(𝑟𝑟𝑒𝑒𝑙𝑙)Ψ𝑁𝑁(𝑟𝑟𝑁𝑁) (A4)

The electronic wave function Ψel can now be expressed as a Hartree product:

Ψ𝑒𝑒𝑙𝑙= 𝜓𝜓1(𝑟𝑟1)𝜓𝜓2(𝑟𝑟2) … 𝜓𝜓𝑛𝑛(𝑟𝑟𝑛𝑛) (A5)

Here the individual one electron wave functions denoted by 𝜓𝜓_𝑖𝑖 can now be called molecular orbitals and they represent the spatial function of the individual electrons. However, the Hartree product wave function is not antisymmetric. Antisymmetry is an important property of all fermions such as electrons and is a consequence of the Pauli Exclusion Principle. To address this condition we can construct an antisymmetrised wave function using Slater determinants:

8. Appendix – Theoretical Methods Ψ =_√𝑛𝑛!1 � 𝜓𝜓1(𝑟𝑟1) 𝜓𝜓2(𝑟𝑟1) 𝜓𝜓1(𝑟𝑟2) 𝜓𝜓2(𝑟𝑟2) ⋯ ⋯ 𝜓𝜓𝜓𝜓𝑛𝑛𝑛𝑛(𝑟𝑟(𝑟𝑟12)) ⋮ ⋮ ⋱ ⋮ 𝜓𝜓1(𝑟𝑟𝑛𝑛) 𝜓𝜓2(𝑟𝑟𝑛𝑛) … 𝜓𝜓𝑛𝑛(𝑟𝑟𝑛𝑛) � (A6)

This antisymmetrised wave function is also called the Hartree wave function and it consists of one electron molecular orbitals. We can then express these one electron molecular orbitals using a linear combination of atomic orbitals or the LCAO approach:

𝜓𝜓𝑖𝑖(𝑟𝑟𝑖𝑖)= ∑ 𝐶𝐶𝜇𝜇 𝜇𝜇𝑖𝑖𝜒𝜒𝜇𝜇(𝑟𝑟𝑖𝑖) (A7)

Here 𝐶𝐶_{𝜇𝜇𝑖𝑖} are the molecular orbital coefficients and 𝜒𝜒_𝜇𝜇 are atomic orbitals or also called basis functions. Using all the aforementioned approximations we have now reduced solving the Schrödinger equation to calculating the molecular orbital coefficients. However, in order to calculate these coefficients we need to know the molecular orbitals, which in turn relies on computing the coefficients to determine the energy. To address this we invoke the Variational Principle which in quantum mechanics states that any approximate wave function is always higher in energy than the exact wave function. Thus the variational principle sets an upper limit to the energy of the exact wave function and iterative minimisation of the energy with respect to the molecular orbital coefficients after an initial guess can be used to obtain the ‘optimised’ molecular orbitals and energies. This is also known as the Self Consistent Field (SCF) method.3–5 Until now, electron spin has been omitted. Each molecular orbital which was previously described as a spatial function also contains a spin component. These spin components are the electron spin eigenfunctions α and β.

Using all of the mentioned approximations we can finally arrive at the corresponding Hartree-Fock equations which have the following form:

𝐹𝐹�𝜒𝜒𝑖𝑖 = 𝜀𝜀𝑖𝑖𝜒𝜒𝑖𝑖 (A8)

Here 𝐹𝐹� is the Fock operator and 𝜀𝜀_𝑖𝑖 is the energy of the spin orbital 𝜒𝜒_𝑖𝑖. The Hartree-Fock equations can be further simplified by the introduction of a basis set of the form in (A7) to be restructured into the matrix form of the Roothan equations:6

FC = SCε (A9)

F is the Fock matrix, C is the matrix of the molecular orbital coefficients, S is the overlap matrix and ε is a diagonal matrix of orbital energies 𝜀𝜀_𝑖𝑖. Since this is now a 159

8. Appendix – Theoretical Methods

generalised eigenvalue problem, it can be solved iteratively as mentioned before where an initial guess of the coefficients C until no change is observed between cycles. Although this approach, also known as Restricted Hartree-Fock (RHF) is reasonable for closed-shell systems with no unpaired electrons as the α and β electrons can be considered energetically degenerate other techniques such as Unrestricted Hartree-Fock (UHF) and Restricted Open-Shell Hartree-Fock (ROHF) are available to address open- shell systems such as free radicals. However, as this thesis does not include any open- shell systems discussion of these techniques are considered outside its scope.