Many-body perturbation theory accuracy

From this, we may deduce that the relativistic correction to the correlation energy is dominated by the contribution from the s electron pair, and that the total relativistic effect involving the exchange of a single transverse Breit photon is obtained to sufficient accuracy for our present purposes at second-order in many-body perturbation theory. 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

The details of SAPT are beyond the scope of the present work. For our purposes it is enough to say that the fundamental components of the interaction energy are ordinarily expanded in terms of two perturbations the intermonomer interaction operator and the intramonomer electron correlation operator. Such a treatment provides us with fundamental components in the form of a double perturbation series, which should be judiciously limited to some low order, which produces a compromise between efficiency and accuracy. The most important corrections for two- and three-body terms in the interaction energy are described in Table 1. The SAPT corrections are directly related to the interaction energy evaluated by the supermolecular approach, Eq.(2), provided that many body perturbation theory (MBPT) is used [19,28]. Assignment of different perturbation and supermolecular energies is shown in Table 1. The power of this approach is its open-ended character. One can thoroughly analyse the role of individual corrections and evaluate them with carefully controlled effort and desired... 

Solution of the matrix equations associated with an independent particle model gives rise to a representation of the spectrum which is an essential ingredient of any correlation treatment. Finite order many-order perturbation theory(82) forms the basis of a method for treating correlation effects which remains tractable even when the large basis sets required to achieve high accuracy are employed. Second-order many-body perturbation theory is a particularly simple and effective approach especially when a direct implementation is employed. The total correlation energy is written... 

S. Wilson and D. Moncrieff, On the accuracy of the algebraic approximation in molecular electronic structure calculations. VI. Matrix Hartree-Fock and Many-Body Perturbation Theory Calculations for the Ground State of the Water Molecule, preprint... 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

The all-orders relativistic many-body perturbation theory approach [82], [83], the combination of this approach with the multiconfiguration Dirac - Fock method [84] or the relativistic coupled-cluster approach [85] allow for the evaluation of the energy levels for valence electrons with accuracy of the order of... 

We now have a well defined prescription for the calculation of the properties of atoms and molecules within a relativistic formulation. As in the non-relativistic case, the relativistic many-body perturbation theory becomes increasingly complicated in higher orders and in practice it is possible to take the expansion to about fourth order with the size of basis set that is required for calculations of useful accuracy. Sapirstein82 has recently re-iterated the view that... 

The applications of many-body perturbation theory in contemporary research in the molecular sciences are manifold and it is certainly not possible to describe more than a mere fraction of the enormous number of publications which have exploited this approach to the molecular structure problem over recent years. Calculations based on second order many-body perturbation theory or MP2 theory are particularly prevalent offering unique advantages in terms of efficiency and accuracy over many other theoretical and computational approaches. Here, we shall briefly describe the use of graphical user interfaces and then concentrate on two recent applications of the many-body perturbation theory which have established new levels of precision. 

Finite basis set Hartree-Fock calculations yield not only an approximation for the occupied orbitals but also a representation of the spectrum which can be used in the treatment of correlation effects. In particular, the use of finite basis sets facilitates the effective evaluation of the sum-over-states which arise in the many-body perturbation theory of electron correlation effects in atoms and molecules. Basis sets have been developed for low order many-body perturbation theoretic treatments of the correlation problem which yield electron correlation energy components approaching the suh-milliHartree level of accuracy [20,21,22]. 

Almost all applications of quantum mechanics to the problem of quantitatively describing molecular electronic structure begin with the choice of a suitable basis set in terms of which the electronic wavefunction is then parametrized. This choice of basis set is crucial, since it ultimately determines the accuracy of the calculation, whether it be a matrix Hartree-Fock calculation, a conflguration-interaction study or a many-body perturbation theory expansion, whether it is a calculation of the total energy of a system, the energy of interaction between two subsystems, or the determination of some electric or magnetic property. 

---