10.6: Beyond H₂⁺ - General MO theory - Hartree-Fock Self-Consistent Field (SCF) theory

By making the independent particle approximation, we wrote the Hamiltonian as a sum of single-electron problems. However, these single-electron wavefunctions completely ignore the fact that electrons repel each other, so the orbitals and energies derived are quite wrong. You can imagine that if two electrons are right next to one another, there will be high potential energy due to repulsion, but our calculation did not take this into account at all. The “right” way to fix the problem is to solve the Hamiltonian with the \(r_{12}\) term included, but no one knows how to do that (if indeed it is even possible!). Since we can’t do that, we’ll refine our approximation to get a better solution, and point to a direction in which we can systematically improve our method until it approaches the true solution.

The approach we’ll use is called Hartree-Fock (HF) Self-Consistent Field (SCF) theory. HF-SCF theory is the simplest electronic structure theory that makes qualitatively accurate predictions about molecular structure, bonding, and energetics of chemical reactions, and it serves as the foundation of nearly all of the more advanced theories that give highly accurate answers. It was originally developed by British physicist Douglas Hartree before it was known that electrons possessed spin, but was later refined by the Russian physicist Vladimir Fock to include spin. The physical idea behind HF-SCF theory is that once a particular electron configuration has been chosen, it is possible to calculate the electron density from the total electronic wavefunction; i.e., the Slater determinant. If the average electron density is known, then the average electron-electron repulsion can be calculated from the expectation value of the \(1/r_{ij}\) operator:

\[E_{e-e} = \sum_{\substack{i,j \\[4pt] i < j}} \left \langle \psi \left | \dfrac{1}{r_{ij}} \right | \psi \right \rangle, \nonumber \]

where \(\ket{\psi}\) is the electronic wavefunction associated with a particular Slater determinant.

It’s highly likely that the shapes of our MOs (as determined by the weight coefficients of each AO) determined by the independent particle approximation are not ideal; once we add in electrons, occupied MOs that overlap with each other in many places would cause the electrons to strongly repel, and we did not take that into account at all. We need to make adjustments to the shapes of the orbitals in an effort to reduce the average repulsion energy. To do so, we add the contribution to the electron-electron repulsion energy from each combination of orbitals to the Hamiltonian matrix that we had diagonalized using the independent particle approximation. Doing so introduces off-diagonal terms, so the resultant matrix (called the Fock matrix) is no longer diagonal. We then diagonalize the Fock matrix to obtain new orbital shapes and energies that take electron repulsion into account. However, the new orbital shapes cause the repulsion energy to change, so we have to recalculate it, yielding a new (non-diagonal) Fock matrix, rediagonalize to obtain new solutions, then recalculate the repulsion energy, and so on. This iterative process is repeated until the difference in repulsion energy (or the magnitude of the largest off-diagonal element of the Fock matrix) from one cycle to the next becomes sufficiently small: we then say that the electron repulsion is self-consistent with the molecular orbitals that give rise to it, and we have achieved a “self-consistent field.” For this reason, the iterative process we just described is called the SCF procedure.

Here, we will outline the procedure for a polyatomic molecule in its ground electronic state (on a problem set, you will examine some of the terms involved in when this procedure is applied to ). In general, the calculations are too involved to do by hand, and many specialized software packages are available for performing them. We will discuss these in somewhat more detail in Section 1.5. For a polyatomic molecule with \(N\) electrons and \(K\) nuclei, the spin-free, non-relativistic, time-independent electronic Hamiltonian (i.e., the Hamiltonian under the Born-Oppenheimer approximation) is:

\[\hat{H} = -\frac{1}{2}\sum_i^N \nabla_i^2 - \sum_{i,k}^{N,K} \frac{Z_k}{r_{ik}} + \sum_{\substack{k,q \\[4pt] q < k}}^{K,K} \frac{Z_kZ_q}{R_{kq}} + \sum_{\substack{i,j \\[4pt] j < i}}^{N,N} \frac{1}{r_{ij}} \nonumber \]

where indices \(i\) and \(j\) run over electrons, and indices \(k\) and \(q\) run over nuclei. Since we’re making the BO approximation, the third term is a constant and can simply be added to the SCF energy after the procedure converges.

Here is the general procedure we follow to calculate the SCF wavefunction and energy:

1. First, break the Hamiltonian into a sum of single electron problems, a perturbation that contains the electron repulsion, and the nuclear potential (which is constant): \[\begin{aligned} \hat{H} = & \sum_i^N\hat{H}^0_i + \hat{H^1} + \sum_{\substack{k,q \\[4pt] q < k}}^{K,K} \frac{Z_kZ_q}{R_{kq}}\\[4pt] \hat{H}_i^0 = & -\frac{1}{2} \nabla_i^2 - \sum_{k}^K \frac{Z_k}{r_{ik}} \\[4pt] \hat{H}^1 = & \sum_{\substack{i,j \\[4pt] j < i}}^{N,N} \frac{1}{r_{ij}} \end{aligned} \nonumber \]
   Here, \(\hat{H}^0\) is an independent electron Hamiltonian (often called the “core Hamiltonian”), \(\hat{H}^1\) has the electron-electron interaction, and the remaining term is the constant nuclear repulsion.
2. Choose a set of basis functions \(\ket{\chi_i}\) to describe the atomic orbitals of all atoms in the molecule, such as hydrogenic wavefunctions centered on each atom. Later we will look at different types of “basis” wavefunctions that are commonly employed in modern quantum chemical calculations. Using this basis, construct the core Hamiltonian matrix under the independent particle approximation; its matrix elements are \(H_{ij} = \bra{\chi_i}\hat{H}_0\ket{\chi_j}\).
3. Diagonalize the single-electron Hamiltonian matrix to calculate LCAO-MOs \(\ket{\phi_i}\) and their energies. The number of MOs that comes out is equal to the number of atomic orbitals \(\ket{\chi_i}\) put in, and they will serve as our initial guess for the shapes of the orbitals in the HF-SCF procedure which treats average electron-electron repulsion.
4. Write the Slater determinant \(\ket{\psi}\) for the ground electronic state and calculate the average electron-electron repulsion \(\bra{\psi}\hat{H}^1\ket{\psi}\). Expressed in terms of the MOs, the electron-electron repulsion interaction can be added to the diagonalized core Hamiltonian matrix by evaluating six-dimensional integrals. After some simplification, for a closed-shell molecule with \(N\) electrons the matrix elements of the electron-electron repulsion interaction are: \[\begin{aligned} H_{ij}^1 & = \sum_{k}^{N/2} 2\tilde{J}_{ik} - \tilde{K}_{kj} \\[4pt] \tilde{J}_{ik} & = \left \langle \phi_i^1\phi_k^2 \left | \dfrac{1}{r_{12}} \right | \phi_i^1\phi_k^2 \right \rangle \\[4pt] \tilde{K}_{kj} & = \left \langle \phi_k^1\phi_j^2 \left | \dfrac{1}{r_{12}} \right | \phi_j^1\phi_k^2 \right \rangle, \end{aligned} \nonumber \] where the summation index \(k\) runs over the occupied molecular orbitals (i.e., the MOs that contain an electron). The \(\tilde{J}_{ik}\) and \(\tilde{K}_{kj}\) integrals are the two-electron Coulomb and exchange integrals, which are conceptually very similar to the one-electron Coulomb and exchange integrals we encountered in the single-electron Hamiltonian. The two-electron Coulomb integral accounts for the classical repulsion between the two “electron clouds” described by each electron’s orbital, and the two-electron exchange integral is a quantum effect arising from the Pauli exclusion principle.
5. The Fock operator \(\hat{f} = \hat{H}^0 + \hat{H}^1\) can then be constructed in matrix form and diagonalized to yield new orbitals and energies. However, with new orbitals, the electron-electron repulsion energy is now different than before, as it depends on the orbital shapes. The Fock operator itself depends on the orbitals used to calculate it. Thus, rather than diagonalizing a matrix and obtaining new MOs, then recalculating, instead it is most common to follow an iterative approach which uses the atomic basis orbitals directly. In the atomic orbital basis \(\ket{\chi_i}\), the HF-SCF procedure iteratively solves the Roothan-Hall equation (shown here for a closed-shell molecule with \(M\) atomic basis orbitals):
   \[\begin{aligned} \hat{F}\mathbf{C} & = \mathbf{SC}\epsilon \\[4pt] F_{ij} & = H_{ij}^{\text{core}} + \sum_a^{N/2} \sum_{\lambda,\sigma}^{M,M}C_{\lambda a}C_{\sigma a}^* \left(2 \left \langle \chi_i^1\chi_\sigma^2 \left | \dfrac{1}{r_{12}} \right | \chi_j^1\chi_\lambda^2 \right \rangle - \left \langle \chi_i^1\chi_\sigma^2 \left | \dfrac{1}{r_{12}} \right | \chi_\lambda^1\chi_j^2 \right \rangle \right),\end{aligned} \nonumber \] where \(\hat{F}\) is the Fock operator, \(\mathbf{C}\) is the matrix of weighting coefficients for the basis functions \(\ket{\chi_i}\), \(\mathbf{S}\) is the overlap matrix, and \(\epsilon\) represents the orbital energies. The summation index \(a\) runs over the occupied MOs, and the \(\lambda\) and \(\sigma\) indices run over all of the basis atomic orbitals. The main difference between the Roothan-Hall equation and the Schödinger equation is that the Fock operator itself depends on the \(C_{mn}\) coefficients. Typically, initial guesses for the \(C_{mn}\) coefficients come from solving the Schrödinger equation for the core Hamiltonian. From there, the procedure proceeds iteratively until the changes in the Fock matrix elements \(F_{ij}\) are very small from one cycle to the next. Once this condition is reached, the self-consistent field is said to have converged, and the orbital energies are self-consistent with the average repulsion energies of the electrons in the orbitals used to generate the Fock matrix.

The total electronic energy is the sum of the single electron energies and the electron-electron interaction energies. For an \(N\)-electron closed shell molecule:

\[E_{\text{elec}} = \sum_i^{N/2} 2E_i + \sum_{i,j}^{N/2,N/2} 2\tilde{J_{ij}} - \tilde{K_{ij}}, \label{chap:mot:eq:fock} \]

where the summations run over all occupied molecular orbitals.

It is important to remember that the HF-SCF procedure is just an approximation: it uses the Born-Oppenheimer approximation, and it treats the interactions between electrons only in an average sense. Moreover, since it is an approach based on the variational principle, the choice of the trial functions also has an impact on the accuracy within the HF-SCF approximation itself. If the electronic energy derived from the SCF procedure are independent of the chosen LCAO basis (i.e., adding more basis AOs does not affect the results), the electronic energy and wavefunction are said to be at the Hartree-Fock limit: they are the best solutions to the Schrödinger equation that can be written as products of single-electron orbitals.

However, even in the Hartree-fock Limit, the solutions are not exact because HF-SCF theory cannot account for dynamic electron correlation. Physically, electrons just tend not to get very close to one another; they “like” to stay apart, so it is said that their positions are “correlated.” In the HF-SCF approximation, the electrons are placed individually into orbitals (limited only by the Pauli exclusion principle), then the average electron-electron interaction energy was calculated based on the orbital shapes. In truth, this process actually overestimates the repulsion because an electron in one orbital is biased against being located close to an electron in another orbital. By calculating the electron positions one-by-one independently of the others, nothing in the HF-SCF approximation prevents them from getting so close together, so the repulsion is anomalously high. Because products of single-electron wavefunctions cannot dynamically account for electron correlation, they will always overestimate the electronic energy: the gap between the Hartree-Fock energy and the “true” electronic energy is called the correlation energy.

HF-SCF theory is the simplest theory that makes qualitatively correct predictions about molecular structure and energetics for many systems. The energies of occupied orbitals in HF-SCF theory are, according to Koopman’s thoerem, equal to the amount of energy required to eject a single electron in that orbital from the molecule. This prediction can be tested using photoelectron spectroscopy, in which molecules are placed in a vacuum chamber and exposed to single-frequency UV light. As the frequency of the UV light is varied, photocurrent from ejected electrons is detected when the energy of a UV photon is equal to an orbital energy. The photoelectron spectrum gives an experimental measure of the orbital energies, and HF-SCF theory is typically accurate to about 1 eV. However, for most modern chemical applications, HF-SCF theory is nowhere near accurate enough to make reliable quantitative predictions. Nevertheless, nearly all of the advanced methods used today take the HF-SCF energy and wavefunctions as a starting point for the calculation.

Under the Born-Oppenheimer approximation, the electronic wavefunction is independent of the nuclear coordinates, but the electronic energy does depend parametrically on the positions of the nuclei. That is, the electronic energy changes if the nuclei are in different locations. By solving the electronic Schrödinger equation at many different nuclear configurations, a potential energy surface is obtained. The minimum energy that we find on the PES is the equilibrium geometry, but as we discussed before there may be multiple local minima on the surface for different conformers/isomers. In addition to minima, there will likely be local maxima indicating transition states and multiple asymptotes representing possible dissociation channels.

Two examples of PESs are shown in Figure 1.9. On the left is an example of a PES that shows the energies of several important conformers, isomers, transition states, and dissociation channels for the \(\ce{HOCO}\) radical. The lines connecting the stationary points and dissociation channels indicate direct pathways through which the nuclei can rearrange without going through any other minima or maxima: these indicate direct paths along a certain reaction coordinate. Figures like this one give a quick overview of the energetics associated with a collection of nuclei in different configurations, and illustrate the relative energies of molecules and the endo/exothermicity of chemical reactions. The right side of Figure 1.9 shows a local part of the PES for the complex along two particular internal coordinates that correspond to torsional motion. A figure like this shows how the energy varies continuously as the molecular geometry varies, so it is possible to see the minimum energy path connecting two particular stationary points.

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