The response electron–electron repulsion energy and energy component analysis in CC/MBPT methods

Abstract

Molecular properties are computed as responses to perturbations (energy derivatives) in coupled-cluster (CC)/many-body perturbation theory (MBPT) models. Here, the CC/MBPT energy derivative with respect to a general two-electron (2-e) perturbation is assembled from gradient theory for 2-e property evaluation, including the electron repulsion energy. The correlation energy (∆E) is shown to be the sum of response kinetic (∆T), electron–nuclear attraction (∆V), and electron repulsion (∆V _ee ) energies. Thus, evaluation of total V _ee for energy component analysis is simple: For total energy (E), total 1-e responses T and V, and nuclear–nuclear repulsion energy (V _NN ), V _ee  = E − V _NN  − T − V is the true 2-e response value. Component energy analysis is illustrated in an assessment of steric repulsion in ethane’s rotational barrier. Earlier SCF-based results (Bader et al. in J Am Chem Soc 112:6530, 1990) are corroborated: The higher-energy eclipsed geometry is favored versus staggered in the two repulsion energies (V _NN and V _ee ), while decisively disfavored in electron–nuclear attraction energy (V). Our best quality calculations (CCSD/cc-pVQZ) attain practical Virial Theorem compliance (i.e., agreement among the kinetic energy, potential energy, and total energy representations) in assigning 2.70 ± 0.06 to the barrier height; −195.80 kcal/mol is assigned to the drop in “steric” repulsion upon going to the eclipsed geometry. Steric repulsion is not responsible for any fraction of the ~3 kcal/mol barrier.

Appendix

First, inspection of each of the 1-e parts of $H_{N}^{A\chi } ,H_{N}^{B\lambda } ,\, {\text{and}}\,H_{N}^{C\gamma }$, [see Eq. (6)] allows us to assign the respective CPHF perturbation matrices for the three individual perturbations:

$$ Q_{pq}^{\chi } = \sum\limits_{k} {\left\langle {pk||qk} \right\rangle } ,\quad Q_{pq}^{\lambda } = T_{pq} ,\quad {\text{and}}\quad Q_{pq}^{\gamma } = V_{pq} . $$

Clearly, their sum is the unperturbed Fock matrix $\left( {f_{pq} } \right)$:

$$ Q_{pq}^{\chi } + Q_{pq}^{\lambda } + Q_{pq}^{\gamma } = f_{pq}^{{}} $$

Second, from CPHF theory and the choice of a particular set of noncanonical SCF orbitals, we expand the derivative of any virtual orbital (a) in terms of occupied orbitals (i) only [14–17]:

$$ a^{\chi } = - \sum\limits_{i} {U_{ai}^{\chi } i = - \sum\limits_{i} {\sum\limits_{bj} {B_{ai,bj} Q_{bj}^{\chi } i} } } ;\quad {\text{where}}\quad B_{ai,bj} = \frac{{A_{ai,bj}^{ - 1} }}{{(\varepsilon_{i} - \varepsilon_{b} )}}\quad {\text{and}}\quad A_{ai,bj} = \delta_{ai,bj} + \frac{{\left\langle {ab||ij} \right\rangle + \left\langle {aj||ib} \right\rangle }}{{(\varepsilon_{a} - \varepsilon_{i} )}} \, . $$

The expansion is fairly simple because there is no displacement of atomic orbitals for this kind of perturbation. Likewise, $a^{\lambda } \;{\text{and}}\;a^{\gamma }$ are expanded. Collecting the sum of the three derivatives:

$$ a^{\chi } + a^{\lambda } + a^{\gamma } = - \sum\limits_{i} {\left( {U_{ai}^{\chi } + U_{ai}^{\lambda } + U_{ai}^{\gamma } } \right)i = - \sum\limits_{i} {\sum\limits_{bj} {B_{ai,bj} \left( {Q_{bj}^{\chi } + Q_{bj}^{\gamma } } \right)} } } i = - \sum\limits_{i} {\sum\limits_{bj} {B_{ai,bj} f_{bj} = 0} } , $$

since the elements of the vir.–occ block of the Fock matrix are all zeroes. A similar argument holds for the derivative of any occupied orbital (i): $i^{\chi } + i^{\lambda } + i^{\gamma } = 0$.