Second Quantization Formalism Research Articles

Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation.

This paper presents a general second-quantized form of a permutation operator interchanging n pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalized form of the permutation operator has enabled the derivation of universal formulas for S2n approximations of the exchange energies in the first and second order of the interaction operator. We present expressions for corrections of S4 approximations and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in V. This new approach offers an alternative to the existing formalism based on density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.

Second quantization of many-body dispersion interactions for chemical and biological systems

The many-body dispersion (MBD) framework is a successful approach for modeling the long-range electronic correlation energy and optical response of systems with thousands of atoms. Inspired by field theory, here we develop a second-quantized MBD formalism (SQ-MBD) that recasts a system of atomic quantum Drude oscillators in a Fock-space representation. SQ-MBD provides: (i) tools for projecting observables (interaction energy, transition multipoles, polarizability tensors) on coarse-grained representations of the atomistic system ranging from single atoms to large structural motifs, (ii) a quantum-information framework to analyze correlations and (non)separability among fragments in a given molecular complex, and (iii) a path toward the applicability of the MBD framework to molecular complexes with even larger number of atoms. The SQ-MBD approach offers conceptual insights into quantum fluctuations in molecular systems and enables direct coupling of collective plasmon-like MBD degrees of freedom with arbitrary environments, providing a tractable computational framework to treat dispersion interactions and polarization response in intricate systems.

Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations.

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.

Energy polarization and energy pumping in Rice-Mele chains

The one-dimensional Rice-Mele lattice consisting of two-site unit cells enables topological charge pumping at low enough temperatures. We show here that energy pumping is feasible in this system. We analyze energy polarization using the second-quantization formalism and connect it to intercell energy current by the relevant continuity equation. Following this formulation, we numerically evaluate the energy current pumped in an adiabatic cycle when the system parameters vary periodically and slowly with time. Unlike the pumped charge, the pumped energy is entirely temperature-independent at half filling. The global Berry phase for all bands and the related topological phase transitions during the pumping cycle account for the energy pumping. The present study paves the way for manipulating energy current without electric and thermal biases.

Chaotic evolution of the energy of the electron orbital and the hopping integral in diatomic molecule cations subjected to harmonic excitation

We analysed the dynamics of the positively charged ions of diatomic molecules (X2+ and XY+), in which the bond is realized by the single electron. We assumed that the atomic cores separated by the distance R were subjected to the external excitation of the harmonic type with the amplitude A and frequency Ω. We found the ground states of ions using the variational approach within the formalism of second quantization (the Wannier function was reproduced by means of Gaussian orbitals). It occurred that, on the account of the highly non-linear dependence of the total energy on R, the chaotic dynamics of cores induced the chaotic evolution of the electronic Hamiltonian parameters (i.e. the energy of the electron orbital ε and the hopping integral t). Changes in cation masses or in the charge arrangement does not affect qualitatively the values of Lyapunov exponents in the A-Ω parameter space.

Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Molecular second-quantized Hamiltonian: Electron correlation and non-adiabatic coupling treated on an equal footing.

We introduce a new theoretical and computational framework for treating molecular quantum mechanics without the Born-Oppenheimer approximation. The molecular wavefunction is represented in a tensor-product space of electronic and vibrational basis functions, with electronic basis chosen to reproduce the mean-field electronic structure at all geometries. We show how to transform the Hamiltonian to a fully second-quantized form with creation/annihilation operators for electronic and vibrational quantum particles, paving the way for polynomial-scaling approximations to the tensor-product space formalism. In addition, we make a proof-of-principle application of the new Ansatz to the vibronic spectrum of C2.

Electromagnetically induced grating with second field quantization in spherical semiconductor quantum dots

A new approach for diffracting the weak probe beam into higher-order directions is proposed via electromagnetically induced grating in second field quantization formalism, offering a new way for implementations of quantum information with semiconductor quantum dots. The formalism of second field quantization allows describing atoms and photons as a many-body system. An induced diffraction grating is formed based on the electromagnetic induced transparency when a standing-wave coupling field is applied to a spherical quantum dot as a three-level system. Due to phase modulation, the zeroth-order light intensity becomes weak, and the first-order diffraction is improved affectedly. On the contrary, the probe beam is barely diffracted via absorption modulation. The simulation results verify that photon numbers of probe and control fields, as well as other parameters in the QD, can lead to the diffraction efficiency of phase grating to be improved. Phase diffraction grating accompanied with a high transmissivity is demonstrated, and the first-order diffraction efficiency reaches $$30\%$$. Also, the impact of QD dimensions on its optical response is investigated. This model may find potential applications in designing the semiconductor quantum dot-based photonic devices in optical communications and quantum information networks.

Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy.

The multiconfiguration time-dependent Hartree (MCTDH) method is a powerful method for solving the time-dependent Schrödinger equation in quantum molecular dynamics. It is, however, hampered by the so-called curse of dimensionality which results in exponential scaling with respect to the number of degrees of freedom in the system and, thus, limits its applicability to small- and medium-sized molecules. To avoid this scaling, we derive equations of motion for a series of truncated MCTDH methods using a many-mode second-quantization formulation where the configuration space is restricted based on mode-combination levels as also done in the vibrational configuration interaction and vibrational coupled cluster methods for solving the time-independent Schrödinger equation. The full MCTDH wave function is invariant with respect to the choice of constraint (or gauge) operators, but restricting the configuration space removes this invariance. We, thus, analyze the remaining redundancies and derive equations for variationally optimizing the non-redundant matrix elements of the constraint operators. As an alternative, we also present a constraint that keeps the density matrices block diagonal during the propagation and the two choices are compared. Example calculations are performed on formyl fluoride and a series of high-dimensional Henon-Heiles potentials. The results show that the MCTDH[n] methods can be applied to large systems and that an optimal choice of constraint operators is key to obtaining the correct physical behavior of the wave function.

Coherence and interactions between geodesics

Following our previous paper, where it was shown that classical internal forces are the outcome of entangled states that collapsed into labeled states, we show that reformulating the GR geodesic equation of motion in a formalism of second quantization causes distant geodesics to interact.

Jastrow-type Decomposition in Quantum Chemistry for Low-Depth Quantum Circuits.

We propose an efficient O(N2)-parameter ansatz that consists of a sequence of exponential operators, each of which is a unitary variant of Neuscamman's cluster Jastrow operator. The ansatz can also be derived as a decomposition of T2 amplitudes of the unitary coupled cluster with generalized singles and doubles, which gives a near full-CI energy. The proposed ansatz therefore can reproduce the uCCGSD energy, or rather will reach the exact full-CI energy because of the exponential operator product form. Because the cluster Jastrow operators are expressed by a product of number operators and the derived Pauli operator products, namely, the Jordan-Wigner strings, are all commutative, it does not require the Trotter approximation to implement to a quantum circuit and should be a good candidate for the variational quantum eigensolver algorithm of a near-term quantum computer. The accuracy of the ansatz was examined for dissociation of a nitrogen dimer, and compared with other existing O(N2)-parameter ansatzs. Not only the original ansatzs defined in the second-quantization form but also their Trotter-splitting variants, in which the cluster amplitudes are optimized to minimize the energy obtained with a few, typically single, Trotter steps, were examined by quantum circuit simulators.

Semidefinite Relaxation of Multimarginal Optimal Transport for Strictly Correlated Electrons in Second Quantization

We consider the strictly correlated electron (SCE) limit of the fermionic quantum many-body problem in the second-quantized formalism. This limit gives rise to a multimarginal optimal transport (MM...

Nonlinear optical response of the α−T3 model due to the nontrivial topology of the band dispersion

We study the electronic contribution to the nonlinear optical response of the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ model. This model is an interpolation between a graphene $(\ensuremath{\alpha}=0)$ and dice $(\ensuremath{\alpha}=1)$ lattice. Using a second-quantized formalism, we calculate the first- and third-order responses for a range of $\ensuremath{\alpha}$ and chemical potential values as well as considering a band gap in the first-order case. Conductivity quantization is observed for the first-order, while higher-order harmonic generation is observed in the third-order response with the chemical potential determining which applied field frequencies both quantization and harmonic generation occur at. We observe a range of experimentally accessible critical fields between ${10}^{2}--{10}^{6}$ V/m with dynamics depending on $\ensuremath{\alpha},\ensuremath{\mu}$, and the applied field frequency. Our results suggest an $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$-like lattice could be an ideal candidate for use in terahertz devices.

Multiconfigurational Self-Consistent Field Theory with Density Matrix Embedding: The Localized Active Space Self-Consistent Field Method.

Density matrix embedding theory (DMET) is a fully quantum-mechanical embedding method which shows great promise as a method of defeating the inherent exponential cost scaling of multiconfigurational wave function-based calculations by breaking large systems into smaller, coupled subsystems. However, we recently [ Pham et al. J. Chem. Theory Comput. 2018 , 14 , 1960 .] encountered evidence that the approximate single-determinantal bath picture inherent to DMET is sometimes problematic when the complete active space self-consistent field (CASSCF) is used as a solver and the method is applied to realistic models of strongly correlated molecules. Here, we show this problem can be defeated by generalizing DMET to use a multiconfigurational wave function as a bath without sacrificing practically attractive features of DMET, such as a second-quantization form of the embedded subsystem Hamiltonian, by dividing the active space into unentangled active subspaces each localized to one fragment. We introduce the term localized active space (LAS) to refer to this kind of wave function. The LAS bath wave function can be obtained by the DMET algorithm itself in a self-consistent manner, and we refer to this approach, introduced here for the first time, as the localized active space self-consistent field (LASSCF) method. LASSCF exploits a modified DMET algorithm, but it is a variational wave function method; it does not require DMET's ambiguous error function minimization, and it reproduces full-molecule CASSCF in cases where comparable DMET calculations fail. Our results for test calculations on the nitrogen double-bond dissociation potential energy curves of several diazene molecules suggest that LASSCF can be an appropriate starting point for a perturbative treatment. Outside of the context of embedding, the LAS wave function is inherently an attractive alternative to a CAS wave function because of its favorable cost scaling, which is exponential only with respect to the size of individual fragment active subspaces, rather than the whole active space of the entire system.

Exponential parameterization of wave functions for quantum dynamics: Time-dependent Hartree in second quantization

We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration interaction wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules.

Energy Levels in a Single-Electron Quantum Dot with Hydrostatic Pressure

In this article we present a study of the effects of hydrostatic pressure on the energy levels of a quantum dot with an electron. A quantum dot is modeled using an infinite potential well and a two-dimensional harmonic oscillator and solved through the formalism of second quantization. A scheme for the implementation of a quantum NOT gate controlled with hydrostatic pressure is proposed.

Nuclear Structure from the In-Medium Similarity Renormalization Group

Efforts to describe nuclear structure and dynamics from first principles have advanced significantly in recent years. Exact methods for light nuclei are now able to include continuum degrees of freedom and treat structure and reactions on the same footing, and multiple approximate, computationally efficient many-body methods have been developed that can be routinely applied for medium-mass nuclei. This has made it possible to confront modern nuclear interactions from Chiral Effective Field Theory, that are rooted in Quantum Chromodynamics with a wealth of experimental data.Here, we discuss one of these efficient new many-body methods, the In-Medium Similarity Renormalization Group (IMSRG), and its applications in modern nuclear structure theory. The IMSRG evolves the nuclear many-body Hamiltonian in second-quantized form through continuous unitary transformations that can be implemented with polynomial computational effort. Through suitably chosen generators, we drive the matrix representation of the Hamiltonian in configuration space to specific shapes, e.g., to implement a decoupling of low-and high-energy scales, or to extract energy eigenvalues for a given nucleus.We present selected results from Multireference IMSRG (MR-IMSRG) calculations of open-shell nuclei, as well as proof-of-principle applications for intrinsically deformed medium-mass nuclei. We discuss the successes and prospects of merging the (MR-)IMSRG with many-body methods ranging from Configuration Interaction to the Density Matrix Renormalization Group, with the goal of achieving an efficient simultaneous description of dynamic and static correlationsin atomic nuclei.

Excitation energies with spin-orbit couplings using equation-of-motion coupled-cluster singles and doubles eigenvectors.

A method of calculation of excited states with spin-orbit couplings, which utilizes left and right eigenvectors of equation-of-motion coupled-cluster singles and doubles model has been formulated and implemented. The spin-orbit interactions are introduced by using the spin-orbit mean field approximation of the Briet-Pauli Hamiltonian. In order to evaluate all the necessary matrix elements, a scheme based on the diagrammatic representation of the second-quantized form of the spin-orbit interaction operator and the standard rules of second-quantized algebra is presented. We posit that this scheme is general and much simpler to use than the often used rules derived for the configuration state functions by using the Wigner-Eckart theorem. We show that the spin-orbit coupled states (i.e., target relativistic states) must satisfy specific conditions in order to classify them according to the double group symmetry. This interrelation between the structure of the target relativistic states and its double group symmetry is discussed in detail. An algorithm to classify the target states according to the irreducible representation of the double group symmetry is offered and implemented. Numerical tests for several atoms and molecules show good agreement of predicted and experimental spin-orbit splittings of the target excited states.

Quantum canonical ensemble: A projection operator approach

Knowing the exact number of particles N, and taking this knowledge into account, the quantum canonical ensemble imposes a constraint on the occupation number operators. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary but fixed N. On the other hand, fixing only the average number of particles, one may remove the above constraint and simply factorize the traces in Fock space into traces over single-particle states. As is well known, that would be the strategy of the grand-canonical ensemble which, however, comes with an additional Lagrange multiplier to impose the average number of particles. The appearance of this multiplier can be avoided by invoking a projection operator that enables a constraint-free computation of the partition function and its derived quantities in the canonical ensemble, at the price of an angular or contour integration.Introduced in the recent past to handle various issues related to particle-number projected statistics, the projection operator approach proves beneficial to a wide variety of problems in condensed matter physics for which the canonical ensemble offers a natural and appropriate environment. In this light, we present a systematic treatment of the canonical ensemble that embeds the projection operator into the formalism of second quantization while explicitly fixing N, the very number of particles rather than the average. Being applicable to both bosonic and fermionic systems in arbitrary dimensions, transparent integral representations are provided for the partition function ZN and the Helmholtz free energy FN as well as for two- and four-point correlation functions. The chemical potential is not a Lagrange multiplier regulating the average particle number but can be extracted from FN+1−FN, as illustrated for a two-dimensional fermion gas.

Normal-ordered second-quantized Hamiltonian for molecular vibrations.

A normal-ordered second-quantized form of the Hamiltonian is derived for quantum dynamics in a bound potential energy surface expressed as a Taylor series in an arbitrary set of orthogonal, delocalized coordinates centered at an arbitrary geometry. The constant, first-, and second-order excitation amplitudes of this Hamiltonian are identified as the ground-state energy, gradients, and frequencies, respectively, of the size-extensive vibrational self-consistent field (XVSCF) method or the self-consistent phonon method. They display the well-defined size dependence of V(1-n/2), where V is the volume and n is the number of coordinates associated with the amplitudes. It is used to rapidly derive the equations of XVSCF and vibrational many-body perturbation methods with the Møller-Plesset partitioning of the Hamiltonian.