Exponential ansatz

However, in the late 1960s, fir Cfzek and Josef Paldus introduced a some what different approach to the electron correlation problem instead of using a linear expansion of functions as in Eq. (13.3) they suggested an exponential ansatz of the general form [7-9]... 

The common exponential ansatz E exp(/ k( no) now is used with a mean or reference refractive index no, and Eq.(22) is transformed to... 

So far, we have specified the wave operator H in the BW form (15). If we adopt an exponential ansatz for the wave operator Cl, we can speak about the single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory. The simplest way how to accomplish the idea of an exponential expansion is to exploit the so-called state universal or Hilbert space exponential ansatz of Jeziorski and Monkhorst [23]... 

If we substitute the Hilbert space exponential ansatz (26) for the wave operator fl, we obtain the system of equations... 

Unfortunately, in contrast to the Cl method, an extension of the SR CC theory to the MR case is far from being straightforward, since there is no unique way in which to generalize the SR exponential Ansatz for the exact N-electron wave function jT), i.e.,... 

We can thus conclude that the complementarity of the Cl and CC approaches in their ability to account, respectively, for the nondynamic and dynamic correlation effects, is worthy of a further pursuit in view of their relative affordability and due to the fact that both types of wave functions are simply related via the exponential Ansatz and yield the same exact result in their respective FCI and FCC limit. 

The single-reference CC theory is based on the exponential ansatz for the ground-state wave function,... 

Jeziorski B, Paldus J (1990) Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function. J Chem Phys 90 2714-2731... 

Using a cluster operator, T, and an exponential ansatz [60,61], the coupled cluster wave function is written as... 

Mukherjee/69/ and Haque and Mukherjee/69/ advocated the use of normal ordered exponential ansatz, in the manner of Lindgren/71/, to avoid the S-S contractions. They showed that C is of the form... 

The cluster expansion methods are based on an excitation operator, which transforms an approximate wave function into the exact one according to the exponential ansatz... 

In this section we examine some of the critical ideas that contribute to most wavefunction-based models of electron correlation, including coupled cluster, configuration interaction, and many-body perturbation theory. We begin with the concept of the cluster function which may be used to include the effects of electron correlation in the wavefunction. Using a formalism in which the cluster functions are constructed by cluster operators acting on a reference determinant, we justify the use of the exponential ansatz of coupled cluster theory. ... 

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... 

It is perhaps useful to compare the exponential ansatz of Eq. [31] with the analogous expansions of other wavefunctions. In the configuration interaction (Cl) approach, for example, a linear excitation operator is used instead of an exponential. 

The exponential ansatz described above is essential to coupled cluster theory, but we do not yet have a recipe for determining the so-called cluster amplitudes (tf. If-- , etc.) that parameterize the power series expansion implicit in Eq. [31]. Naturally, the starting point for this analysis is the electronic Schrodinger equation,... 

The projective techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes that define the coupled cluster wavefunction, e o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, T, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression ... 

The other possibility is to focus on the MR CISD wave function and exploit the Tj(0) and r20) clusters it provides to account for the dynamic correlation due to disconnected triples and quadruples that are absent in the MR CISD wave function. This approach, recently proposed and tested by Meissner and Gra-bowski [42], may thus be characterized as a CC-ansatz-based Davidson-type correction to MR CISD. The duplication of contributions from higher-than-doubly excited configurations that arise in MR CISD as well as through the CC exponential ansatz is avoided by a suitable projection onto the orthogonal complement to the MR CISD N-electron space. The results are very encouraging, particularly in view of their affordability, though somewhat inferior to RMR CCSD. 

The similarity transformed EOM, STEOM-CC [183], approaches the problem somewhat differently, but it also provides an exponential ansatz for excited states, namely = exp(5) exp(7)l0), where exp(5) has a different meaning than before. The method decouples the contributions of higher cluster operators from the lower ones, by using the results for the (1,0) and (0,1) results to define the second similarity transformation, 5, leaving the excited states to be obtained now from a problem of the dimension of a Cl singles (CIS) calculation. This method is a kind of exact CIS for the excited states of molecules, at least those dominated by single excitations. It is very attractive for large-scale application as in our work for free-base porphine [184,185]. Extensions by Nooijen and Lotrich have been made for doubly excited states [186]. 

Two sets of wave-like operators are defined and expanded in coupled-cluster normal-ordered exponential ansatze. Q, = l-Hx is a standard wave operator... 

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