Multireference Function Research Articles

Molecular Application of a State Specific Multi-Reference Brillouin–Wigner Perturbation Theory**Supported by the Scientific and Technological Research Council of Turkey (TUBİTAK) under Grant No 2219-1/2013.

The single reference second order Brillouin–Wigner perturbation theory recently developed, which eliminates its size-extensivity error, has been generalized to state-specific, multi-reference (SS-MR), BWPT2 providing a size-extensive correction to the electron correlation problem for systems that demand the use of a multi-reference function. Illustrative numerical tests of the size-extensivity corrections are made for widely used molecules in their ground states, which are pronounced multi-reference characteristics. We have implemented two-reference and three-reference cases for CH2, BH and bond breaking process in the ground states of HF molecules. The results are compared with the rigorously size-extensive methods such as the Møller–Plesset perturbation theory, i.e., MP2, full configuration interaction (Full-CI) and allied methods using the same basis sets.

An alternative approximation to state-specific multireference second-order Brillouin-Wigner perturbation theory: size-extensivity correction

State-specific, multireference, second order, Brillouin-Wigner perturbation theory is examined provided that providing a size-extensive correction to the electron correlation problem for systems that demand the use of a multireference function. We derived the renormalization term. An important aspect of this renormalization term is to cancel the non-physical terms. The computer implementations are made for bond breaking process in the ground states of the diatomic hydride using the two configuration state functions. We have achieved improved results. The results are compared with the rigorously size-extensive methods such as, MOller-Plesset perturbation theory, that is, MP2, full configuration interaction and allied methods using the same basis sets.

Many‐body Brillouin–Wigner second‐order perturbation theory: A robust and efficient approach to the multireference correlation problem

AbstractThe multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

An orbital-invariant coupled electron-pair like approximant to a state-specific multi-reference coupled cluster formalism

Almost all the coupled electron pair approximation (CEPA)-like developments based on multi-reference functions do not satisfy invariance with respect to separate unitary transformation among doubly occupied, active and virtual orbitals. Those, which do satisfy this criterion, use empirical CEPA-like corrections involving suitably averaged pair-correlation terms. We develop in this Letter a CEPA-like approximant to the state-specific multi-reference coupled cluster (SS-MRCC) formalism [J. Chem. Phys. 100, (1999) 6171] which includes all the exclusion principle violating terms rigorously, and display manifest orbital-invariance. The theory, termed as SS-MRCEPA(I), is also rigorously size-extensive, size-consistent and bypasses intruders in a natural manner.

Alternatives to multireference methods for the molecular electronic structure problem

AbstractThe development of robust many‐body methods for the molecular electronic structure problem with respect to multireference functions has attracted much attention over the past two decades. In recent years, multireference methods based on the Brillouin–Wigner expansion have been shown to overcome the intruder state problems which have plagued similar approaches based on the Rayleigh–Schrödinger series. However, there are some problems in molecular electronic structure theory, which may be handled by means of multireference methods, but can be treated by methods that avoid the use of multireference functions. We consider two examples of alternatives to multireference methods for the molecular electronic structure problem. The study of excited states having the same symmetry as the ground state or some lower lying excited state often involves the use of a multireference function. We describe an alternative procedure based on the generalized Rayleigh–Ritz variation principle. Gidopoulos, Glushkov, and Wilson have termed this the optimized trace method. The generalized Rayleigh–Ritz principle for the relativistic formulation of the molecular electronic structure problem is briefly also considered. Molecular dissociative processes are frequently described by quantum chemical methods based on multireference functions. Single‐reference Hartree–Fock methods usually provide a qualitatively incorrect description of such processes even in the simplest of molecules, H2. However, approximate single configuration wave functions constructed from nonorthogonal orbitals can often afford a useful approximation to bond‐breaking processes. Each occupied nonorthogonal orbital is then an eigenfunction of a different Fock‐like operator. Each of these Fock‐like operators supports a spectrum of M single particle states, where M is the size of the basis set, and only the lowest of these is occupied. We briefly consider the relativistic formulation of the molecular electronic structure problem based on approximations involving products of nonorthogonal functions within the Furry bound state interaction picture of quantum electrodynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Extraction of temporal information in functional MRI

The temporal resolution of functional MRI (fMRI) is limited by the shape of the haemodynamic response function (hrf) and the vascular architecture underlying the activated regions. Typically, the temporal resolution of fMRI is on the order of 1 s. We have developed a new data processing approach to extract temporal information on a pixel-by-pixel basis at the level of 100 ms from fMRI data. Instead of correlating or fitting the time-course of each pixel to a single reference function, which is the common practice in fMRI, we correlate each pixel's time-course to a series of reference functions that are shifted with respect to each other by 100 ms. The reference function yielding the highest correlation coefficient for a pixel is then used as a time marker for that pixel. A Monte Carlo simulation and experimental study of this approach were performed to estimate the temporal resolution as a function of signal-to-noise ratio (SNR) in the time-course of a pixel. Assuming a known and stationary hrf, the simulation and experimental studies suggest a lower limit in the temporal resolution of approximately 100 ms at an SNR of 3. The multireference function approach was also applied to extract timing information from an event-related motor movement study where the subjects flexed a finger on cue. The event was repeated 19 times with the event's presentation staggered to yield an approximately 100-ms temporal sampling of the haemodynamic response over the entire presentation cycle. The timing differences among different regions of the brain activated by the motor task were clearly visualized and quantified by this method. The results suggest that it is possible to achieve a temporal resolution of /spl sim/200 ms in practice with this approach.

Multireference Møller—Plesset perturbation theory for high-spin open-shell systems

New forms of multireference Møller—Plesset perturbation theory are presented that are applicable to high-spin open-shell systems. The second-order perturbation theory based on the multireference function yields large parts of dynamical correlation corrections for a relatively low computational cost. Results are reported for the energy difference between the 1A 1 and 3B 1 states of CH 2 and for the potential energy curve of the triplet ground state ( 3Σ − g) of O 2. In spite of the drastic simplification, results are encouraging.

A cluster expansion theory with multireference functions using the unitary ansatz

A cluster expansion theory with multireference functions is developed. The method is applied to the ground state of H 2, the lowest two 1∑ + states of LiH, and the lowest three 1A 1 states of H 2O. Well-balanced potential curves are obtained for H 2 and LiH. Excitation energies are in good agreement with experiment.