Exponentially correlated wavefunctions

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

Bases of fully exponentially correlated wavefunctions [1, 2] provide more rapid convergence as a function of expansion length than any other type of basis thus far employed for quantum mechanical computations on Coulomb systems consisting of four particles or less. This feature makes it attractive to use such bases to construct ultra-compact expansions which exhibit reasonable accuracy while maintaining a practical capability to visualize the salient features of the wavefunction. For this purpose, exponentially correlated functions have advantages over related expansions of Hylleraas type [3], in which the individual-term explicit correlation is limited to pre-exponential powers of various interparticle distances (genetically denoted r,j). The general features of the exponentially correlated expansions are well illustrated for three-body systems by our work on He and its isoelectronic ions, for... 

The wavefunction described with the optimum a,y corresponds well with a somewhat perturbed lslj 2s configuration. Two of the electrons depend on the electron-nuclear distance with a values (screening parameters) that correspond to a partially screened interaction with the - -3-charged Li nucleus in a split-shell electron distribution. The third electron (that with pre-multiplying r) has an a value somewhat larger than for a hydrogenic 2s orbital, indicative of the fact that the inner-shell electrons do not completely shield the Li nucleus. The electron-electron a values all reflect the existence of electron-electron repulsion, with the effect most pronounced for the Is-ls interaction. All these observations are consistent with the notion that the exponentially correlated wavefunction gives an excellent zero-order description of the electronic structure of Li. 

We can distinguish between two broad classes of explicitly correlated wavefunctions polynomials in r(J and other inter-body distances, and exponential or Jastrow forms [15,16]... 

Energy Computation for Exponentially Correlated Four-Body Wavefunctions... 

There have now been several applications reported for fiilly exponentially correlated four-body wavefunctions [7-9], also limited to bases without pre-exponential r,j. While it was found that pre-exponential are relatively unimportant for three-body systems, they can be expected to contribute in a major way to the efficiency of expansions for three-electron systems such as the Li atom and its isoelectronic ions, as is obvious from the fact that the zero-order description of the ground states of such systems has electron configuration s 2s. 

A practical reason that pre-exponential r,y have not been used with exponentially correlated four-body wavefunctions has been the difficulty of managing analytical formulas for the integrals that thereby result that difficulty has now been reduced in importance by the author s recent presentation[10] of a recursive procedure for the integral generation. 

We present now an illustrative calculation for the 5" electronic ground state of the Li atom, using a single-term spatial wavefunction that can be characterized as an exponentially correlated l Slater-type orbital (STO) product. For the purpose of this illustration, we restrict the spin state (the three-electron doublet space has dimension two) to that in which the Is and are singlet-coupled. Thus, the space-spin wavefunction has the form... 

CC Coupled-cluster theory, where the correlated wavefunction is created from Fg by acting on the latter with an exponential operator exp(T), which creates excited configurations. An infinite-order variant of (->) MBPT. Unlike (—>) MP2 or (- ) Cl, the solution of the CC equations requires an iterative procedure. 

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... 

Exponential parameters were optimized at the SCF level, at the singleconfiguration spin-coupled (SC) level, and for a valence- and core-correlated OBS-GMCSC wavefunction that included eight configurations. Intermediate GMCSC wavefunctions, including three, six and seven configurations, used the exponential parameters that had been optimized at the SC level. 

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... 

The failing of MBPT is that it is basically an order-by-order perturbation approach. For difficult correlation problems it is frequently necessary to go to high orders. This will be the case particularly when the single determinant reference function offers a poor approximation for the state of interest, as illustrated by the foregoing examples at 2.0 R. A practical solution to this problem is coupled-cluster (CC) theory. In fact, CC theory simplifies the whole concept of extensive methods and the linked-diagram theorem into one very simple statement the exponential wavefunction ansatz. 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

Eully correlated exponential wavefunctions for four-body systems... 

Table 1 Computed energies of Li ground state (non-relativistic, Coulomb interaction only, infinite-mass nucleus) for various wavefunctions, in Hartree atomic units. This research is for a correlated exponential premultiplied by the electron-nuclear distance for the 2s electron, with the parameters given in Table 33...

The spectral representations above are not computationally efficient, as they would require knowledge of all intermediate excited states. Computationally tractable formulas for the response functions within the various approximate methods are obtained instead through the following steps (1) choose a time-independent reference wavefunction (2) choose a parametrization of its time-development, for instance an exponential parametrization (3) set up the appropriate equations for the time development of the chosen wavefunction parameters (4) solve these equations in orders of the perturbation to obtain the wavefunction (parameters) (5) insert the solutions of these equations into the expectation value expression and obtain the RTFs and (6) identify the excited-state properties from the poles and residues. The computationally tractable formulas for the response functions therefore differ depending on the electronic structure method at hand, and the true spectral representations given above are only valid in the limit of a frill-configuration interaction (FCI) wavefunction. For approximate methods (i.e., where electron correlation is only partially included), matrix equations appear instead of the SOS expressions, for example. 

As described in Sec.3.1, standard wavepacket propagation schemes can be employed for the evaluation of fiux correlation functions. These methods employ multi-dimensional grids or basis sets to represent the wavefunction. Thus, the numerical effort of these schemes increases exponentially with the number of degrees of freedom. Given the computational resources presently available, only systems with up to four atoms can be treated accurately. The extension of numerically exact calculations towards larger systems therefore requires other schemes for the solution of the time-dependent or time-independent Schrodinger equation. 

Coupled cluster with singles and doubles excitations (CCSD) is a size-consistent post-HF electron correlation method. The wavefunction, Y, in coupled cluster theory is formulated in terms of a cluster (exponential) expansion including the single and double excitation operators 7i and %. The effect of triple excitations (T) is calculated with perturbation theory. 

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