Frozen-orbital approximation excitation energies

Excitation energies calculated with the RPA and TDA approaches for N2 with a moderately large basis set are listed in Table 23. Both the RPA and TDA excitation energies are significantly lower than those obtained with the simplest frozen orbital approximation. All these approaches differ only in their treatment of the final state, and the pattern of predicted excitation energies shows this in a rather dramatic way. Both the RPA and TDA allow for a limited amount of relaxation and provide much improved predictions. Inclusion of the... 

The composition of this review is as follows Section 2 describes the numerical examples of the rules for degenerate excitations. The data in the next section are obtained by highly correlated methods, since the effects of electron correlations are essential for accurate descriptions of the excited states. Section 3 demonstrates the interpretation of the rules by using the simplified model that corresponds to the frozen-orbital approximation (FZOA) [4]. In the excitation energy formulas to which the FZOA leads, the splitting schemes are related to the specific two-electron integrals, whose values are qualitatively analyzed by the relevant orbital characters. Finally, the summary is addressed in Section 4. 

The acronyms representing the algebraic expressions will be explained below. The first term (ORX), which is always negative, arises from the presence of single excitations a r in the second-order energy of the N — l)-particle system. Because of Brillouin s theorem, such excitations would not have contributed if we had used the HF orbitals of the (N — l)-particle system. Their effect is to optimize or relax the orbitals we did use (i.e., the HF orbitals of the iV-particle system) and hence this term is said to arise from orbital relaxation (ORX). Since it lowers the energy of the (N — l)-particle system relative to the frozen orbital approximation, it decreases the Koopmans theorem IP. Consider the second term (PRX), which is also always negative. 

HyperChem supports MP2 (second order Mdllcr-l Icsset) correlation energy calcu latiou s u sin g any available basis set. lu order to save main memory and disk space, the HyperChem MP2 electron correlation calculation normally uses a so called frozen-core approximation, i.e. th e in n er sh el I (core) orbitals are omitted. A sett in g in CHHM.IX I allows excitation s from th e core orbitals to be include if necessary (melted core). Only the single poin t calcula-tion is available for this option. 

The structural parameters and vibrational frequencies of three selected examples, namely, H2O, O2F2, and B2H6, are summarized in Tables 5.6.1 to 5.6.3, respectively. Experimental results are also included for easy comparison. In each table, the structural parameters are optimized at ten theoretical levels, ranging from the fairly routine HF/6-31G(d) to the relatively sophisticated QCISD(T)/6-31G(d). In passing, it is noted that, in the last six correlation methods employed, CISD(FC), CCSD(FC),..., QCISD(T)(FC), FC denotes the frozen core approximation. In this approximation, only the correlation energy associated with the valence electrons is calculated. In other words, excitations out of the inner shell (core) orbitals of the molecule are not considered. The basis of this approximation is that the most significant chemical changes occur in the valence orbitals and the core orbitals remain essentially intact. On... 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

A similar approach can be applied for the Y atom insertion into the C-H bond of alkenes and other alkanes. Our calculation by the Cl method in a 6-311-H-G(2d,2p) basis set with a complete active space for 8 electrons in 8 orbitals (Is orbital of carbon atom is frozen) predicts that the vertical S-T excitation energy in methane is around 11 eV (11.37 eV or 262 kcal/mol). Following the above approximation it is equal -2 Jch- From Eq. (9) the activation energy for the yttrium atom insertion reaction, Eq. (1) M=Y, should be 17.6 kcal/mol. This simple estimation is in a good agreement with very accurate ab initio calculations, Ea = 20.7 kcal/mol [15]. 

The interelectronic energy of an electron in orbital i with two paired electrons in orbital / consists of two parts Jij for the different-spin interaction and Jy — Xy for the same-spin interaction, which together give 2 Jt] — Kij. Within the orbital i only Jn should appear but this term, due to relation (14), may be replaced by 2Ju —Ku. It is important to realize that this self-adjustment occurs only for occupied orbitals — thanks to the property (/< — K ) y)i = 0 — but not for virtual orbitals since for those the operator 2J —Kf is present, and an electron in a virtual orbital feels the full interaction of N electrons. For this reason it is often said that virtual-orbital solutions of (5) are appropriate for (N 4- l)-electron systems It would be natural to use an operator Hke (3) to obtain appropriate virtual orbitals for N-electron systems. This heis been done by Kelly in his extensive perturbation calculation of Be 29-65) by Hunt and Goddard in their calculation of the excited states of H2O >, and by Lefebvre-Brion et al. (frozen-core approximation) Goddard s method will serve to illustrate this general type of treatment. 

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