Reference wavefunction

One fomis a reference wavefunction T (this can be of the SCF, MPn, CC, etc variety) tlie energy differences are computed relative to the energy of this fiinction. 

A different approach for calculating excited states is based on indirect methods that allow one to calculate excitation energies based on a single reference wavefunction. Single reference methods for the calcualtion of excited states of large molecules have... 

A Simple Method for the Evaluation of the Second-Order Perturbation Energy from External Double-Excitations with a CASSCF Reference Wavefunction. 

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

Several of the methods referred to in this chapter use the URCCSD(T) procedure in which a spin-unrestricted CCSD(T) calculation is performed on a high-spin RHF reference wavefunction, as implemented in the MOLPRO program. H. J. Werner, P. J. Knowles, R. D. Amos, A. Bemhardsson, A. Beming, P. Celani, D. 

The transformation U incorporates the additional correlations when going from the reference wavefunction to the target wavefunction t. In the second picture, we appear to have a different Hamiltonian H and eigenstate to-... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

The simplest and most widely-employed method is the so-called configuration interaction singles or CIS method. This involves singleelectron promotions only (from occupied molecular orbitals in the reference wavefunction to unoccupied molecular orbitals). Because there are relatively few of these, CIS is in fact practical for molecules of moderate complexity. As noted previously, single-electron promotions do not lead to improvement in either the ground-state wavefunction or energy over the corresponding Hartree-Fock... 

Here then is the crux of the computational difficulty. The reactant, (3Z)-3-hexene-l,5-diyne, is well described by a single-configuration reference wavefunction. The product, p-benzyne, is likely to have appreciable diradical character and necessitates a multiconfiguration wavefunction. The transition state will exist somewhere in between. The choice of computational method suited to describe all three structures equally well is nontrivial, and in the next section we discuss the various approaches employed and results obtained by a number of research groups. 

The singlet state will require at least two configurations since the NBMOs (MO 22 and 23) are close in energy. Thus, a reasonable reference wavefunction for... 

This size inconsistency occurs because the two open-shell electrons on the atoms must be singlet-coupled to produce the correct dissociation limit, and a supermolecule, two-determinant approach is therefore required. This difficulty also applies to coupled cluster or perturbation-based wavefunctions that use the RHF determinant as a reference these methods cannot be size consistent for a given molecular system unless the reference wavefunction is size consistent. 

Note that f is at most a two-particle operator and that T is at least a one-particle excitation operator. Then, assuming that the reference wavefunction is a single determinant constructed from a set of one-electron functions. Slater s rules state that matrix elements of the Hamiltonian between determinants that differ by more than two orbitals are zero. Thus, the fourth term on the left-hand side of Eq. [48] contains, at the least, threefold excitations, and, as a result, that matrix element (and all higher order elements) necessarily vanish. The energy equation then simplifies to... 

Because such diagrams cannot represent matrix elements that have the reference wavefunction on the right, only the third diagram above can contribute to... 

The choice of as the zeroth-order Hamiltonian requires the use of either a spin-restricted (closed-shell) Hartree-Fock (RHF) or spin-unrestricted Hartree-Fock (UHF) determinant as the zeroth-order (reference) wavefunction. Since spin-restricted open-shell Hartree-Fock (ROHF) reference functions are not eigenfunctions of the spin-orbital P, other partitionings are required (Refs. 127-134). 

Although a spin-orbital formulation is conceptually simple, desirable properties such as spin-adaptation may be lost when the electronic state of interest is open shell, for example. A rigorously spin-adapted theory must include spin-free definitions of the cluster operators, T, and an appropriate (perhaps multideterminant) reference wavefunction (Refs. 39, 41, 42, 156-158). Such general coupled cluster derivations are beyond the scope of this chapter, though some of the issues associated with difficult open-shell problems are discussed in the next section. 

Several researchers have recently devoted considerable effort to the derivation and efficient implementation of techniques based on spin-restricted reference determinants that reduce the computational discrepancy between closed- and open-shell systems. " This emphasis on spin-restricted techniques has resulted in part from a bias toward reference wavefunctions that maintain the spin symmetry of the exact wavefunction (such as the ROHF determinant), but also because of the possible efficiency advantages of spin-restricted methods over unrestricted techniques. Thus, since the component molecular orbitals are constrained to have identical spatial parts for each spin function, it should be possible to construct the correlated wavefunction in a manner that takes advantage of this symmetry. 

We emphasize that the present discussion focuses only on high-spin open-shell systems to which a single-determinant reference wavefunction is applicable. Coupled cluster techniques for low-spin cases, such as open-shell singlets, have been pursued in the literature for many years, however, and provide a fertile area of research (Refs. 158, 167-170). 

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