Origin of Virtual Molecular Orbitals in Hartree-Fock equations

Question

Let's look at scheme

Do I understand correctly, the origin of Virtual Orbitals is a solution of (for, exampre, restricted) HF equation eigenfuncions?

Answer 1

Yes, they result from the HF equations just like the occupied orbitals. The diagram should also be show them as present in the initial set of equations, unless the basis (used for the LCAO) was changed between the two states shown. The number of occupied orbitals plus the number of virtual orbitals is equal to the basis set size because they are the eigensolutions of $$ \mathbf{F}\mathbf{C} = \mathbf{\epsilon}\mathbf{S}\mathbf{C}, $$ where $\mathbf{F}$ is the Fock matrix, $\mathbf{C}$ is the matrix of orbital coefficients, $\mathbf{\epsilon}$ are the eigenvalues which are the orbital energies, and $\mathbf{S}$ is the atomic orbital overlap matrix (because the basis set used is typically not orthogonal). The dimension of all matrices involved is determined by the size of the basis set, which will be finite because otherwise the computational effort becomes infinite.

Answer 2

Let's start with HF equations (we omit the procedure for their derivation), and obtain the pseudo eigenwertproblem: \begin{equation}\label{Fock}\tag{Fock} \hat{F}_i\phi_i' = \varepsilon_i\phi_i'. \end{equation}

The number of such equations is equal to the number of electrons $N$ (we do not consider such methods as RHF, ROHF in general), so $i = 1 \ldots N$ and $\phi' = \{\phi_1, \phi_2, \ldots, \phi_N \}$ This equation is actually only well defined for the orbitals that give the lowest energy.

The next step to find an approximation to actually solve these still pretty complicated systems.

In this approach we map the solutions $\phi_i$ in the form of linear combination of some basis functions: \begin{equation}\label{fLCAO}\tag{fLCAO} \phi_i'(\mathbf{x}) = \sum_{a=1}^{\infty} c_{a,i}\chi_a(\mathbf{x}). \end{equation} The number of such basis functions is infinite.

But in reality, taking into account the savings in computer computing power, the number of basis functions should be reduced to $M$, thus, we already approximate: \begin{equation}\label{LCAO}\tag{LCAO} \phi_i'(\mathbf{x}) = \sum_{a=1}^{M} c_{a,i}\chi_a(\mathbf{x}). \end{equation}

The basis functions are known to us, but only the coefficients in the expansion remain unknown. The problem of solving equations is reduced to a purely algebraic one. How are these coefficients found?

Substitute \ref{LCAO} into \ref{Fock} and multiply left by $ \left\langle\chi_a\right| $: \begin{equation}\label{} \sum_{b = 1}^M c_{ib} \langle\chi_a|\hat{F}_i|\chi_b\rangle = \varepsilon_i \sum_{b=1}^M c_{ib} \langle\chi_a|\chi_b\rangle. \end{equation} or in a more expanded form \begin{equation}\label{sys}\tag{sys} \begin{cases} c_{i,1} (F_{11} - \varepsilon_i S_{11}) + c_{i,2} (F_{12} - \varepsilon_i S_{12}) + \ldots + c_{i,M} (F_{1M} - \varepsilon_i S_{1M}) = 0, \\ c_{i,1} (F_{21} - \varepsilon_i S_{21}) + c_{i,2} (F_{22} - \varepsilon_i S_{22}) + \ldots + c_{i,M} (F_{2M} - \varepsilon_i S_{2M}) = 0, \\ \vdots \\ c_{i,1} (F_{M1} - \varepsilon_i S_{M1}) + c_{i,2} (F_{M2} - \varepsilon_i S_{M2}) +\ldots + c_{i,M} (F_{MM} - \varepsilon_i S_{MM}) = 0. \end{cases} \end{equation}

The coefficients $ c_{ib} $, the number of which is $ M $, will now be unknown. But we still have $ \varepsilon_i $ unknown. To find them, we must use the fact that non-trivial roots must meet the condition: \begin{equation}\label{} \left| \begin{matrix} F_{11} - \varepsilon_i S_{11} & F_{12} - \varepsilon_i S_{12} & \cdots & F_{1M} - \varepsilon_i S_{1M} \\ F_{21} - \varepsilon_i S_{21} & F_{22} - \varepsilon_i S_{22} & \cdots & F_{2M} - \varepsilon_i S_{2M} \\ \vdots & \vdots & \ddots & \vdots \\ F_{M1} - \varepsilon_i S_{M1} & F_{M2} - \varepsilon_i S_{M2} & \cdots & F_{MM} - \varepsilon_i S_{MM} \end{matrix} \right| = 0. \end{equation}

This equation is $ M $ -th degree with respect to $ \varepsilon_i $, so there must be solutions to $ M $ of$ \varepsilon = \{\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_M \} $. Substituting one of these values ​​into the equation \eqref{sys} (for example $ \varepsilon_2 $), we obtain as a solution the coefficients $ \{c_{21}, c_{22}, \ldots, c_{2M} \} $ .

And knowing these coefficients, from the formula \eqref{LCAO} we get the orbital $ \phi_2 $. Similarly, we get all $ M $ orbitals $$\phi' = \{\phi_1, \phi_2, \ldots, \phi_N, \phi_{N + 1}, \ldots, \phi_M \}.$$

Next is the self-consisting procedure, which improves coefitients.

That is, if we had $ N $ electrons, and at the same time had to have $ N $ orbitals, then by introducing $ M $ pieces of basic functions, we expanded the number of orbitals to $ M $, and hence some of these orbitals ($ M-N $ ) will not be occupied. These orbitals are called virtual.

From these orbitals, one can construct a Slater determinant by discarding the $ M-N $ virtual orbitals, thus we have

$$\phi' = \{\phi_1, \phi_2, \ldots, \phi_N \}$$

When approximating a bounded basis by $M$ functions, it is difficult to give a physical meaning to virtual orbitals $\phi'_{virtual} = \{\phi_{N + 1}, \phi_{N + 2}, \ldots, \phi_M \}$. But if we choose the expansion in the form of \eqref{fLCAO}, then theoretically, they would mean the real excited states of the system (see Configuration interaction).