統計的摂動論の開発とモデル空間量子モンテカルロ法における摂動補正に関する研究

概要

Obtaining accurate solutions of the many-body Schrddinger equation while maintaining moderate computational costs has been one of the main challenges especially in condensed matter physics and quantum chemistiy for decades. The exact many-body wave function can be, in principle obtained by the diagonalization of the full configuration interaction (FCI) matrix in a given Hilbert space. However, the dimension of the problem grows combinatorially with the size of the system, which limits the applicability of the method to small number of conelated electrons. Certainly, exact diagonalization is not an option when one wants to treat medium-sized molecules. As a solution, selected configuration interaction (SCI) methods were proposed long time ago, and has been used for a guide function with improved nodes in diffusion Monte Carlo. SCI has renewed attention and has been improved in recent years. Other modem quantum techniques, such as for example density matrix renormalization group (DMRG), have been developed to tackle problems in large Hilbert spaces that were out of reach in the past. Recently, a full coupled-cluster reduction (FCCR) approach has been introduced to demonstrate that selected coupled-cluster is promising for a balanced treatment of dynamic and nondynamic correlation effects on the same footing.

Another approach of importance in recent years is FCI quantum Monte Carlo (FCIQMC) that aims at solving the Schrddinger equation by employing a walker distribution to represent the CI coefocients in a determinantal basis. The imaginary-time evolution (ITE) of the wave function is simulated by QMC comprising of spawning, diagonal death-cloning, and annihilation events. Model space QMC (MSQMC) is an extension of FCIQMC to calculate excited states utilizing effective Hamiltonian formalism.

The Fermion sign problem is mitigated by annihilating walkers with different signs on the same determinants. The sign problem becomes very sever at low walker occupations as the system size increases. This is because determinants with few walkers with fluctuating signs can accidentally give birth to progeny walkers that cause sign-incoherent noise.

The initiator approach has been introduced to overcome this difficulty, achieved by a restriction in the spawning step using initiator determinants with occupation exceeding a prescribed threshold, i.e. a progeny spawned from a non-initiator determinant cannot survive if it turns out to be on an unoccupied determinant The initiator approach enables a wide range of applications, but yet is essentially a truncation in the Hilbert space and it leads to size-inconsistency enors. In a recent study, with the introduction of second-order Epstein-Nesbet corrections the initiator error was improved in a natural and inexpensive manner.

A stochastic algorithm of many body perturbation theory (MBPT) in arbitrary order has been discussed in the literature already. Although deriving the algebraic expressions of higher order perturbation energies is tedious, stochastic algorithms provide convenient ways for their estimates. These stochastic algorithms are reminiscent of the higher-order MBPT algorithm based on FCI. Nevertheless, the main obstacle in practical calculations is that the number of walkers explodes for higher order terms. A truncation in configuration space can potentially bridle this explosive growth of the walkers. How­ ever, as well-known truncated CI methods provide size-inconsistent energies as higher order excitations are necessary for the cancellation of unlinked terms in the perturbative expansion.

In chapter 2 I acquaint the reader with the convenience of Antisymmetrized Goldstone Diagrams (ASG) that will be used to express certain important terms in the perturbation series. For the sake of easy comprehension, I limit my discourses to the ITE of the tiiird order wave function and explain the size-inconsistency problem of the.foxirth order perturbative energy.

When all the perturbative terms are sampled the results are free from flaws and size-consistent energies are obtained in every order. On the contrary, vdien four fold excitations are excluded from the sampling unpleasant imperfections appear and it would be advantageous to sample a different type of perturbation series that is free from such enors. The research of the current chapter focuses on this problem and proposes a stochastic algorithm based on the linked-cluster expansion to provide size-consistent energies in every order. With some modifications in the original algorithm it is possible to implement the sampling of the linearized coupled cluster (LCC) wave function.

In addition to the implementation I performed several calculations using our new methodology to demonstrate the correctness and the usability of stochastic perturbation theory. One of the most intriguing questions was how the accuracy changes with the excitation level of the sampled space. Accuracy in this case might be loosely defined and I mainly focus on the size-inconsistency problem to compare the CI and the LCC expansion.

I also investigate a semi-stochastic algorithm, x^iich is radically different from the previous works. It reduces stochastic noise by using a deterministic subspace, and treating the complementary space using walker dynamics. I express the first order wave function using perturbation theory to be treated deterministically as the fragment represented by this term thought to constitute a very important part of the total wave function. In chapter 2, I derive the necessary formulations for the variation of the residual wave function and provide a discourse on how to implement the individual terms based on the original MSQMC sampling algorithm.

In the numerical results section of chapter 2,1 demonstrate how the standard deviation changes when the residual sampling is applied and I also present calculation results on molecular systems that were previously investigated using stochastic perturbation theory.

In chapter 3,1 start by providing a'brief overview of Epstein-Nesbet perturbation theory and I present a derivation to obtain the formula for , the third order correction of the initiator error to further improve the accuracy of the results. With the calculation of second and third order perturbative corrections one can obtain energies of near FCI accuracy without having to increase the number of walkers during simulation. In the numerical results section, the accuracy and applicability of these conections are demonstrated for several molecular systems. I calculate i-MSQMC energies on the potential energy curve of the nitrogen molecule in several points and I compare my results obtained using second and third order perturbation with MP2 and CCSD(T). Moreover, I investigate the convergence of both systematic and random errors in the calculations and propose a new methodology to mitigate the stochastic noise in the calculations.

In the calculation results section of chapter 3 I shed light on the accuracy of the i-MSQMC energy and the perturbative corrections with respect to the initiator threshold for two molecular systems. Neither of these molecules can be regarded as strongly correlated systems therefore the corrections were expected to exhibit high accuracies. In both cases overcorrelation due to the second order energy was observed, however calculating the third order energy and adding it to obtain the final energy enabled me to obtain near FCI values even at remarkably low walker populations (or with high initiator threshold equivalently).

The description of electronic states at non-equilibrium geometries and bond stretching in quantum chemistry is a nontrivial challenge for all approximate conelated methods. The N2 molecule in equilibrium state possesses a triple bond and the molecule has been under investigation for many years. In chapter 3, I present calculation results for the potential energy curve of the nitrogen molecule in cc-pVDZ basis as well.I also investigate the size-inconsistency of the perturbative corrections and present some calculation data for the singlet-triplet gap of the polyacene series using the second and the third order energy.

In chapter 3,1 finalize my results and draw a general conclusion of my findings.