Excitation operator reference contribution

From (13.1.11) it is apparent that a coupled-cluster wave function - generated, for example, by all possible single- and double-excitation operators - contains contributions from all determinants entering the FCl wave function (although the number of free parameters is usually much smaller). In practice, therefore, we cannot work with the coupled-cluster state in the expanded form (13.1.11) but we must instead retain the wave function in the more compact form (13.1.7), avoiding references to the individual determinants. 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

The equivalent Tamm-Dancoff approximation, for which also 0rhf) represents the reference state, is obtained, if first all terms involving a state transfer operator are omitted, while those that include excitation operators are retained. Therefore, only the elements An and Bn of the generalised Hessian matrix (126) survive. Second, the contribution of jBh, which gives rise to terms that involve double excitations with respect to the RHF reference state, is omitted. Here again, the explicit construction of the intermediate states can be avoided. 

Notice that the dipole-moment operator is considered to be independent of nuclear coordinates (Condon approximation) and spin, and thus it only appears in the integral containing the orbital part of the electronic wavefunction. In (1.5), the summation in the third term is made over all the transitions between vibrational levels of the ground and excited states that contribute to the intensity of the electronic transition. For common molecular systems in the ground electronic state at room temperature, only transitions of the type Xj,o Xf (thereafter referred to as 0 n) need to be considered in most cases. 

Ozone can be destroyed thermally, by electron impact, by reaction with oxygen atoms, and by reaction with electronically and vibrationaHy excited oxygen molecules (90). Rate constants for these reactions are given ia References 11 and 93. Processes involving ions such as 0/, 0/, 0 , 0 , and 0/ are of minor importance. The reaction O3 + 0( P) — 2 O2, is exothermic and can contribute significantly to heat evolution. Efftcientiy cooled ozone generators with typical short residence times (seconds) can operate near ambient temperature where thermal decomposition is small. 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

We refer to our two papers [60,73] for more detailed discussions and for the demonstration of the avoidance of intruders. Here we just emphasize that, to avoid intruders, it would be necessary to choose the IMS appropriately. The special IMS such as QCMS, where one usually puts quasi-degenerate orbitals in one class, and the nondegenerate orbitals in a different class are very appropriate for separability. There are other desirable simplifications also if we work in the QCMS. For the QCMS, the quasiopen operators cannot lead to excitations into the QCMS itself. This automatically separates the contributions of the quasi-open and the closed operators by simply using the projectors exp(T )/ exp(—T ) and exp(T )P exp(—T ), respectively. This is quite convenient for the practical applications. For the QCMS, we thus need to project Eq. (55) with exp(T )l<)), ) <)), lexp(—T ) to get every quasi-open operator present in for each and project onto exp(T )P exp(—T ) to generate W. Eq. (60) are all trivially zero for a QCMS. 

In the second-quantized operators (31) and (32), the summation over the particle indices i,j,... runs over all the electron states of the (complete one-electron) spectrum. If these operators act to the right upon the reference state, i.e. the many-electron vacuum of the particle-hole formalism, some of these (strings of) creation and annihilation operators create excitations while other gives simply zero, i.e. no contribution. For the pure vacuum, in particular, the behavior of the second-quantized operators can be read off quite easily because the creation operators appear left of the annihilation operators in expressions (31) and (32), respectively. 

This suggests that the intermediate virtual states in these perturbation terms are two particle one hole (2p-h) or two holes one particle (2h-p) excitations out of the reference state. Since the rings and ladders are important contributions to the self-energy, one can assume that restricting the complement space to the operators... 

---