Anharmonic vibrational many-body methods are developed for and applied to small molecules and extended systems in a bound potential energy surface (PES). Diagrammatically size-consistent and basis-set-free vibrational coupled-cluster (XVCC) theory for both zero-point energies and transition frequencies, the latter through the equation-of-motion (EOM) formalism, is defined for an $n$th-order Taylor-series PES. Quantum-field-theoretical tools (the rules of normal-ordered second quantization and Feynman--Goldstone diagrams) for deriving their working equations are established. The equations of XVCC and EOM-XVCC are derived and implemented with the aid of computer algebra. Algorithm optimizations known as strength reduction, intermediate reuse, and factorization are carried out before code generation,producing algorithms with optimal cost scaling. A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. An extended STEOM-XVCC (Ext-STEOM-XVCC) method is defined as an $m$th order configuration interaction method with the doubly similarity-transformed Hamiltonian including up to $m$th-order excitation operators. Because the doubly transformed Hamiltonian is dressed with higher-order anharmonic effects, the frequencies of overtones and combinations obtained are different and superior to the corresponding EOM-XVCC method. We compare and contrast the Ext-STEOM-XVCC method to its electronic counterpart. We apply the previously established second-order size-extensive vibrational many-body perturbation (XVMP2) method to the anharmonic phonon dispersion curves of a model two-mass system and the optical phonons of polyethylene. We obtain accurate results despite the presence of multiple Fermi-resonances in the crystalline systems.
Anharmonic vibrational many-body methods are developed for and applied to small molecules and extended systems in a bound potential energy surface (PES). Diagrammatically size-consistent and basis-set-free vibrational coupled-cluster (XVCC) theory for both zero-point energies and transition frequencies, the latter through the equation-of-motion (EOM) formalism, is defined for an nth-order Taylor-series PES. Quantum-field-theoretical tools (the rules of normal-ordered second quantization and Feynman--Goldstone diagrams) for deriving their working equations are established. The equations of XVCC and EOM-XVCC are derived and implemented with the aid of computer algebra. Algorithm optimizations known as strength reduction, intermediate reuse, and factorization are carried out before code generation,producing algorithms with optimal cost scaling. A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. An extended STEOM-XVCC (Ext-STEOM-XVCC) method is defined as an mth order configuration interaction method with the doubly similarity-transformed Hamiltonian including up to mth-order excitation operators. Because the doubly transformed Hamiltonian is dressed with higher-order anharmonic effects, the frequencies of overtones and combinations obtained are different and superior to the corresponding EOM-XVCC method. We compare and contrast the Ext-STEOM-XVCC method to its electronic counterpart. We apply the previously established second-order size-extensive vibrational many-body perturbation (XVMP2) method to the anharmonic phonon dispersion curves of a model two-mass system and the optical phonons of polyethylene. We obtain accurate results despite the presence of multiple Fermi-resonances in the crystalline systems.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.
High Resolution Image Download MS PowerPoint Slide We present a novel formulation of the vibrational density matrix renormalization group (vDMRG) algorithm tailored to strongly anharmonic molecules described by general, high-dimensional model representations of potential energy surfaces. For this purpose, we extend the vDMRG framework to support vibrational Hamiltonians expressed in the so-called n -mode second-quantization formalism. The resulting n -mode vDMRG method offers full flexibility with respect to both the functional form of the PES and the choice of the single-particle basis set. We leverage this framework to apply, for the first time, vDMRG based on an anharmonic modal basis set optimized with the vibrational self-consistent field algorithm on an on-the-fly constructed PES. We also extend the n -mode vDMRG framework to include excited-state-targeting algorithms in order to efficiently calculate anharmonic transition frequencies. We demonstrate the capabilities of our novel n -mode vDMRG framework for methyloxirane, a challenging molecule with 24 coupled vibrational modes.
Coupled cluster (CC) theory is the “gold standard” of molecular electronic structure theory, yet the full arc of the subject—from its variational origins in Hartree–Fock theory, through the exponential ansatz and its diagrammatic machinery, to its modern reformulation as a system of polynomial equations on an algebraic variety, its multireference generalizations, and its emerging reincarnation as a quantum circuit—is rarely presented as a single connected narrative. This review builds that narrative completely from first principles and carries it through to today’s research frontier. Starting from the electronic Schrödinger equation, we derive Hartree–Fock theory, second quantization, and Wick’s theorem; construct the coupled cluster singles-and-doubles (CCSD) equations via the similarity-transformed Hamiltonian; and show how these same equations reappear, verbatim, as a truncated eigenvalue problem on a projective variety generalizing the Grassmannian, with a computable CC degree controlling the number of solutions. We then develop the perturbative triples correction CCSD(T), the linear-response equationof-motion (EOM-CC) framework for excited, ionized, and electron-attached states, the effective-Hamiltonian formalism underlying multireference coupled cluster theory and its notorious intruder-state problem, and the unitary coupled cluster (UCC) ansatz that restores the variational principle at the cost of an infinite operator expansion—tamed, on a quantum computer, by Trotterization and the variational quantum eigensolver. We close with a synthesis of what is rigorously known about the resulting nonlinear equations, a map of directions deliberately left out of scope, and an assessment of the open question of quantum advantage in this domain. Every claim is stated formally, with a full statement and proof or proof sketch, and cited to its origin; the computational cost of every method is compared explicitly throughout.
Scope The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT) , particularly in its diagrammatic representation (referred to as many-body perturbation theory , or MBPT ), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods. The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).
A size-extensive multidimensional many-body theory is developed from an order-expanded van Vleck transformation. This provides an effective Hamiltonian in a model space consisting of a set of determinants whose zeroth-order energies may be nondegenerate. Expressions for the effective Hamiltonian in terms of the perturbation and a set of resolvents generalized from the Rayleigh–Schrödinger form are given. Perturbative evaluation of the resultant formulas via diagrammatic expansion is illustrated and discussed. The diagrams required through second order for a model space consisting of a Hartree–Fock solution plus selected singly and doubly excited determinants are presented, and their relation to those employed in the method of Hose and Kaldor is discussed.
This thesis deals with selected aspects of a new implementation of many-body methods which can be formulated in the framework of second quantization, in particular the coupled-cluster (CC) method with arbitrary excitation level. Coupled-cluster is one of the most successful and widely used quantum chemical methods for accurate calculations on small to medium-sized molecules. Since it employs a nonlinear parametrization of the wave function, its implementation is rather difficult, in particular if higher (i.e. more than double) excitations are to be included. The latter is necessary to obtain highly accurate results and also as a prerequisite for the generalization to multi-reference cases. The implementation described here has a twofold focus. One is on generality and flexibility regarding the method to be implemented, the other is on efficiency. To achieve flexibility, it is useful to have a machinery which automatically derives working equations for a given method. We realize this by applying techniques of second quantization. This work treats in particular the last step of this procedure, namely the simplification of the resulting equations by the identification of equivalent terms. The algorithm used here is based on the interpretation of algebraic terms as graphs. The derived CC equations then have to be solved iteratively. The efficiency of the program is mainly determined by the evaluation of the occurring expressions, which has to be done in each iteration step. The evaluation is split up in a sequence of tensor contractions. Their generic implementation is complicated by the particular structure of the involved tensors. We reduce each contraction to a sequence of matrix multiplications, which requires a previous data rearranging. But since matrix multiplication is the most efficient operation on modern computers, this additional effort pays off. Preliminary tests show that our program is at least as fast as the most efficient general coupled-cluster implementation so far, and the relation is expected to improve for calculations with larger basis sets where the matrix multiplication becomes the time-determining step. Finally, we give an outlook to possible further developments. In particular, the preparation of the equations before the actual evaluation (factorization) offers much potential for optimizations which we do not exploit at the moment, in contrast to the program with which we compare, which employs at least a partial optimization at this point.