The Frozen Core Approximation

It is quite common in applications of the CI method to invoke the frozen core   approximation, in which the lowest-lying molecular orbitals (occupied by the inner-shell electrons) are constrained to remain doubly-occupied in all configurations. The frozen core for atoms lithium to neon typically consists of the 1s atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals 1s, 2s, 2p$_{\rm x}$, 2p$_{\rm y}$ and 2p$_{\rm z}$. The frozen molecular orbitals are those which are primarily these inner-shell atomic orbitals (or linear combinations thereof).

A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than are the valence electrons. Thus the error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms. In fact, it is sometimes recommended that one employ the frozen core approximation as a general rule because most of the basis sets commonly used in quantum chemical calculations do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons.

Not only does the frozen core approximation reduce the number of configurations in the CI procedure, but it also reduces the computational effort required to evaluate matrix elements between the configurations which remain. Assuming that all frozen core orbitals are doubly occupied and orthogonal to all other molecular orbitals, then it can be shown [21] that

$$\langle \Phi_I \vert \hat{H} \vert \Phi_J \rangle = \langle \bar{\Phi}_I \vert \hat{H}_0 \vert \bar{\Phi}_J \rangle$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred18_I | H | _J = ¯_I | H_0 | ¯_J )

where $\bar{\Phi}_I$ and $\bar{\Phi}_J$ are identical to $\Phi_I$ and $\Phi_J$, respectively, except that the core orbitals have been deleted from $\bar{\Phi}_I$ and $\bar{\Phi}_J$, and $\hat{H}$ has been replaced by $\hat{H}_0$ defined by

$$\hat{H}_0 = E_c + \sum_{i=1}^{N-N_c} \hat{h}_c(i) + \sum_{i>j}^{N-N_c} \frac{1}{r_{ij}}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred19H_0 = E_c + _i=1^N-N_c h_c(i) + _i>j^N-N_c 1r_ij )

where N is the number of electrons and N_c is the number of core electrons. E_c is the so-called ``frozen-core energy,''   which is the expectation value of the determinant formed from only the N_c core electrons doubly occupying the n_c = N_c / 2 core orbitals

$$E_c = 2 \sum_i^{n_c} h_{ii} + \sum_{ij}^{n_c} \left\{ 2 (ii\vert jj) - (ij\vert ji) \right\}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred20E_c = 2 _i^n_c h_ii + _ij^n_c { 2 (ii|jj) - (ij|ji) } )

Finally, $\hat{h}_c(i)$ is the one-electron Hamiltonian operator for electron i in the average field produced by the N_c core electrons,

$$\hat{h}_c(i) = \hat{h}(i) + \sum_{j=1}^{n_c} \left\{ 2 \hat{J}_j(i) - \hat{K}_j(i) \right\}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred21h_c(i) = h(i) + _j=1^n_c { 2 J_j(i) - K_j(i) } )

with $\hat{J}_j(i)$ and $\hat{K}_j(i)$ representing the standard Coulomb and exchange operators, respectively. Note that, although we have written the frozen core energy E_c and frozen core operator $\hat{h}_c$ in terms of molecular orbitals, it is not necessary to explicitly transform the one- and two-electron integrals involving core orbitals. Assuming real orbitals, we can define a frozen core density matrix [22] in atomic (or symmetry) orbitals as

$$P^c_{\rho \sigma} = \sum_i^{n_c} C_{\rho}^{i} C_{\sigma}^{i}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred22P^c_ = _i^n_c C_^i C_^i )

where $C_{\rho}^{i}$ is the contribution of atomic orbital $\rho$to molecular orbital i. Now the frozen core operator in atomic orbitals becomes

$$h^{c}_{\mu \nu} = h_{\mu \nu} + 2 \sum_{\rho \sigma} (\rho \s... ..._{\rho \sigma} (\rho \mu \vert \nu \sigma) P^{c}_{\rho \sigma}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred23h^c_ = h_ + 2 _ (| ) P^c_ - _ (| ) P^c_ )

and the frozen core operator in molecular orbitals h^c_ij can be obtained simply by transforming $h^{c}_{\mu \nu}$. Similarly the frozen core energy can be evaluated as

$$\sum_{\mu \nu} P^{c}_{\mu \nu} \left( h_{\mu \nu} + h^c_{\mu \nu} \right)$$ tex2html_deferredtex2html_deferred4.tex2html_deferred24E_c = (tex2html_deferredtex2html_deferred4.tex2html_deferred24E_c &=& _ P^c_ ( h_ + h^c_ ) )

= Tr(P^c h) + Tr(P^c h^c)

 

An analogous approximation is the deleted virtual   approximation, whereby a few of the highest-lying virtual (unoccupied) molecular orbitals are constrained to remain unoccupied in all configurations. Since these orbitals can never be occupied, they can be removed from the CI procedure entirely because no terms involving them contribute to the CI coefficients or energy. The rationalization for this procedure is that it is unlikely that electrons will choose to partially populate high-energy orbitals in their attempt to avoid other electrons. However, this conclusion is generally true only for very high-lying virtual orbitals (such as those formed by antisymmetric combinations of symmetry orbitals in the core region). For all other virtual orbitals, such simplistic reasoning is not sufficient.

Davidson points out that those high energy SCF virtual orbitals which result from the antisymmetric combination of the two basis functions describing each atomic orbital in a double-$\zeta$ basis set (such as the 3p-like orbital formed from the minus combination of the larger and smaller 2p atomic orbitals on oxygen) often make the largest contribution to the correlation energy in Møller-Plesset (MPn) wavefunctions [23]. This can be seen from the expression for the second-order correction to the energy in Møller-Plesset perturbation theory,

$$E^{(2)} = \sum_{a>b} \sum_{r>s} \frac{\vert\langle ab \vert ... ...e\vert^2} {\epsilon_a + \epsilon_b - \epsilon_r - \epsilon_s}$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred25 E^(2) = _a>b _r>s |ab | | rs |^2 _a + _b - _r - _s )

where $\epsilon_i$ is the orbital energy (eigenvalue) of orbital i. Thus a virtual orbital r with a large energy $\epsilon_r$ will contribute to a large energy denominator in each term of equation (4.25) in which it appears. However, if orbital r lies close spatially to one of the orbitals a or b occupied in the reference, then this large overlap will contribute to a large two-electron integral $\langle ab \vert \vert rs \rangle$. This integral is squared in the numerator, leading to a large overall contribution to the second-order energy. For the antisymmetric combinations of the basis functions describing the core region, this large numerator is insufficient to overcome the even larger energy denominator; such virtual orbitals can generally be deleted with minimal loss in the correlation energy recovered.

Although the analysis in the preceeding paragraph is based on perturbation theory, similar conclusions can be drawn for the CI method. This is most easily verified by actual calculations, since analytical expressions for the energetic contribution of orbitals to the CI energy are not nearly as simple to obtain or interpret as is equation (4.25).