Expectation values coupled-cluster theory

It is possible for groups of three or more lines to be identified as equivalent, though this can happen only in many-body perturbation theory, expectation value coupled cluster theory, or unitary coupled cluster theory. For such diagrams, a prefactor of where n is the number of electron lines, must be included. 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

T. Korona, B. Jeziorski, One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory. 

Tables III-XVII give calculated permanent moments. Selected comparisons with experimental values or calculations of others are also listed. All values are in atomic units, and traceless rather than Cartesian forms are distinguished with Greek letters, 6 (quadrupole) and G (octupole). Coordinates for the atomic centers are listed. These specify the geometry used, which were equilibrium geometries, and implicitly the multipole expansion center (x = 0, y = 0, z = 0). The moments are given at both the SCF level and at the well-correlated level of coupled-cluster theory [95-102]. ACCD [103-106] was the particular coupled-cluster approach, and the moments were evaluated by expectation [102] with the cluster expansion truncated at single and double substitutions.

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

In the case of the variational methods, SCF, MCSCF and CI, o0 = l o) and we have the normal expectation value. For the non-variational methods such as Mpller-Plesset perturbation or coupled cluster theory, the energy is calculated as a transition expectation value, where I FoO =... 

The additive separability of each commutator leads to a formulation of coupled-cluster theory where each term (i.e. each expectation-value expression) in the energy or in the amplitude equations is separately size-extensive. The linked equations are therefore said to he termwise size-extensive. No terms that violate the size-extensivity arise and no cancellation of such terms ever occurs. 

Equation (13.5.8), which represents the generalization of the Hellmann-Feynman theorem to coupled-cluster wave functions, is shown in Exercise 13.1 to give size-extensive first-order properties. For variational wave functions, the Hellmann-Feynman theorem contains the real average value of the operator V (4.2.51) rather than a transition expectation value as in (13.5.8). Likewise, to ensure teal properties, we may in coupled-cluster theory work in terms of the manifestly real expression... 

However, although spin-restricted coupled-cluster theory does not provide a spin-adapted wave function, it does afford a description where the expectation value of the spin operator is exact. 

The last method used in this study is CCSD linear response theory [37]. The frequency-dependent polarizabilities are again identified from the time evolution of the corresponding moments. However, in CCSD response theory the moments are calculated as transition expectation values between the coupled cluster state l cc(O) and a dual state... 

In contrast to variational methods, perturbation theory and coupled-cluster methods achieve their energies from a transition formula < I H I F > rather than from an expectation value... 

This result is easily generalized the normal-ordered form of an operator is simply the operator itself minus its reference expectation value. For the Hamiltonian example, above, the normal-ordered Hamiltonian is just the Hamiltonian minus the SCF energy (i.e., may be considered to be a correlation operator). Owing to its considerable convenience for coupled cluster and many-body perturbation theory analyses, this conventional form of f given in Eq. [105] is adopted for the remainder of this chapter. 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

Non-variational methods such as Mpller-Plesset perturbation theory (MP) or the coupled cluster (CC) method, where the energy can be expressed as a transition or asymmetric expectation value... 

In non-variational approaches such as Moller-Plesset perturbation theory or coupled cluster methods the wavefunction is not at all variationally optimized. However, it is possible to choose ( o hi such a way that the Hellmann-Feynman theorem is fulfilled to a certain extent, while the transition expectation value in Eq. (9.88) still gives the energy. 

Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include MoUer-Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with non-iterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. 

The coupled-cluster (CC) theory for a molecular solute described within the PCM model exploits the concept of the coupled-duster energy functional. For isolated molecules this energy functional corresponds to the coupled-cluster expectation value of the molecular hamUtonian operator (Bartlett and Musial 2007). The corresponding functional for molecular solute has an... 

Given the product ansatz for the coupled-cluster wave function (13.1.7), let us consider its optimization. We recall that, in Cl theory, the wave function (13.1.8) is determined by minimizing the expectation value of the Hamiltonian with respect to the linear expansion coefficients ... 

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