The exponential ansatz

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,... 

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 ... 

One of the main advantages of the Monte Carlo method of integration is that one can use any computable trial function, including those going beyond the traditional sum of one-body orbital products (i.e., linear combination of Slater determinants). Even the exponential ansatz of the coupled cluster (CC) method [27, 28], which includes an infinite number of terms, is not very efficient because its convergence in the basis set remains very slow. In this section we review recent progress in construction and optimization of the trial wavefunctions. 

By means of the exponential ansatz = exp(T), the Bloch equation leads directly to the coupled-cluster approach. For general open-shell systems, it is often convenient to use the normal-ordered exponential [15]... 

Diagrammatically, these cluster operators S,- represent the connected /-body terms, i.e. those diagrams which cannot be separated into topologically unconnected parts. Typically, the wave and cluster operators are related to each other due to the exponential ansatz... 

Myers CR, Umrigar CJ, Sethna JP, Morgan JD III (1991) Fock s expansion, Kato s cusp conditions, and the exponential ansatz. Phys Rev A 44 5537-5546... 

Taking the exponential ansatz for the wave operator (60), the Bloch-like equation (41) then becomes... 

The above representation of the exact ground-state wave function is reminiscent of the exponential ansatz of the single-reference CC theory, Eq. (3). There is, however, a fundamental difference between Eqs. (124) and (3). The cluster operator T entering Eq. (3) is defined in terms of the excitation operators Eq. (6), where ii,..., in (ai,..., an) are the... 

Note that the expansion is the same as the exponential ansatz up to the second-order term, so that for a low-order approximation it makes no difference which we use. 

None of the transformations we have considered will transform this operator completely into an even operator. If we use the exponential ansatz of (16.23), the first term in the... 

The coupled-cluster wave function is defined by the exponential ansatz [15-18]... 

Specification of the coupled-cluster wavefunction and the corresponding energy requires determination of the so far unknown amplitudes tf, tf, ... which are used to parametrize the exponential ansatz in equation (1). The usual procedure for this - often referred to as the standard coupled-cluster approach - involves the following steps... 

Unlike the Cl equations (equations 18 and 19), the CC equations (equations 7 and 8) are nonlinear due to the exponential ansatz and, thus, more complex. Advantages of the CC approach are only apparent when approximations in form of truncation of the excitation operators T and C are introduced. Such approximations are dictated by practical considerations, as untruncated calculations (i.e., FCI) are only feasible for a few small molecules. 

The use of the exponential ansatz in formulating a quantum mechanical many-body theory - a theory which is extensive and scales linearly with the number of electrons studied - was first realized in nuclear physics by Coester and Kiimmel [64-66], The origins of the cluster approach to many-fermion systems, goes back to the early 1950s, when the first attempts were made to understand the correlation effect in an electron gas [67,68] and in nuclear matter [69], For both of these systems, it was absolutely essential that the method employed scaled linearly with the number of particles, i.e. that it is size extensive . 

N is the total number of electrons in the system. The advantage of the exponential ansatz for the wave operator can be seen by considering the application to systems consisting of independent subsystems. A, B, C,. The cluster ansatz for the supersystem considered at the same excitation level can be written as the product of exponential cluster ansatze truncated at that excitation level, that is... 

Now we shall use simple arguments based on our previous results for many-body perturbation theory to show how the exponential ansatz of the coupled cluster method arises. Let us recall eq. (3.251) which gives the exact energy in the form... 

The use of the exponential ansatz in formulating a quantum mechanical many-body theory was briefly described in Chapter 3, Section 3.3.2. This approach was first realized in nuclear physics by Coester and Ktimmel [81,82] and its introduction into quantum chemistry is usually attributed to Click. [76]. A recent overview of this method has been given by Paldus [73]. The single-reference coupled cluster approach has been described [83] as... 

One of the possible ways of realizing the wave operator f2o is to use the exponential ansatz introduced into atomic and molecular electronic structure theory by Cf ek [76] in coupled pair many-electron theory and coupled cluster theory. Here the wave operator is written as... 

If we use the exponential ansatz defined in (4.26) for wave operator f o, we can obtain the following equations for the exact ground state energy ... 

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