Strong orthogonality is an important constraint placed on geminal wavefunctions in order to make variational minimization tractable. However, strong orthogonality prevents certain, possibly important, excited configurations from contributing to the ground state description of chemical systems. The presented method lifts strong orthogonality constraint from geminal wavefunction by computing a perturbative-like correction to each geminal independently from the corrections to all other geminals. The method is applied to the Singlet-type Strongly orthogonal Geminals variant of the geminal wavefunction. Comparisons of this new SS(p)G method are made to the non-orthogonal AP1roG and the unconstrained Geminal Mean-Field Configuration Interaction method using small atomic and molecular systems. The correction is also compared to Density Matrix Renormalization Group calculations performed on long polyene chains in order to assess its scalability and applicability to large strongly correlated systems. The results of these comparisons demonstrate that although the perturbative correction is small, it may be a necessary first step in the systematic improvement of any strongly orthogonal geminal method.
Strong orthogonality is an important constraint placed on geminal wavefunctions in order to make variational minimization tractable. However, strong orthogonality prevents certain, possibly important, excited configurations from contributing to the ground state description of chemical systems. The presented method lifts strong orthogonality constraint from geminal wavefunction by computing a perturbative-like correction to each geminal independently from the corrections to all other geminals. The method is applied to the Singlet-type Strongly orthogonal Geminals variant of the geminal wavefunction. Comparisons of this new SS(p)G method are made to the non-orthogonal AP1roG and the unconstrained Geminal Mean-Field Configuration Interaction method using small atomic and molecular systems. The correction is also compared to Density Matrix Renormalization Group calculations performed on long polyene chains in order to assess its scalability and applicability to large strongly correlated systems. The results of these comparisons demonstrate that although the perturbative correction is small, it may be a necessary first step in the systematic improvement of any strongly orthogonal geminal method.
The Lewis electron-pair picture pervades all of chemistry and a substantialdomain of materials science. “Geminals” are electron-pair quantum states whichare used to express approximate solutions of the Schrödinger equation of polyelec-tronic systems. In quantum chemistry, geminal-based methods are potentially moreeffective than the traditional ones based on the orbital picture: electrons occupyorbitals and experience only the average effect of each other. This is especially truefor strongly correlated-systems. However, the computational cost of using arbitrarygeminal wave functions is non-polynomial in the system size.It is therefore an important challenge to find additional constraints on geminalmodels which preserve the accuracy of the wavefunction description while reducingthe calculations to polynomial cost. The Ayers’ group at Mc Master has developedone such successful geminal model called AP1roG because the wavefunction is anantisymmetric product of geminals with one distinct reference orbital occupied ineach geminal [1]. This ansatz relaxes the so-called “strong-orthogonality” constraintbetween geminals. The latter makes wavefunction matrix element computationsvery easy but appears too drastic from the physical point of view, since the geminalelectron pairs are then distinguishable.We have followed a different path to relax the strong-orthogonality constraint. Wehave defined the geometrical concept of n-orthogonality [2] which can be viewed asa graded indistinguishability measure for electronic states: 1-orthogonality coincideswith strong orthogonality, the larger n the less distinguishable the n-orthogonal elec-tronic states will be. We have studied geminal models constrained by n-orthogonalityrelations for different values of n, imposed to different combinations of geminal prod-uct states [3]. However, in this work the scaling of matrix element computations wasstill unsatisfactory. We will present new ideas developed since then to remedy thisproblem.References[1] P. A Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, J. of Chem. Theory and Computation 9, 1394-1401 (2013).[2] P. Cassam-Chenaı̈, Phys. Rev. A77, 032103 (2008).[3] P. Cassam-Chenaı̈, V. Rassolov, Chem. Phys. Lett. 487, 147-152 (2010).
A new truncation scheme for non-orthogonal antisymmetrized products of interacting geminals (APIG) is introduced based on antisymmetrized products of strongly orthogonal geminals (APSG). This wavefunction hierarchy of interacting geminals (IG) allows us to gradually increase the accuracy at which the ground state of a seniority-zero Hamiltonian can be estimated, ranging from APSG up to approximation-free APIG. Mathematical expressions for the lowest four orders (IG0, IG1, IG2, and IG3) are given explicitly and the computational cost to evaluate their transition density matrix is verified to scale only cubically with system size. Exemplary numerical calculations indicate that already a very early truncation level leads to results virtually identical to APIG.
A new family of geminal Ansätze called antisymmetrized product of 2D-block geminals (AP2D-BG) has been introduced recently to solve the electronic Schrödinger equation of molecules. They build on an antisymmetrized product of strongly orthogonal geminals (APSG) while lifting the "strong orthogonality" and "seniority zero" restrictions. The combinatorics to select the excitations which can be added to each geminal of the product grows exponentially with the number of geminals. However, in this study on diatomic molecules, we present simple guidelines to drastically reduce the number of excitations, which are useful to lower the electronic energy in variational calculations. Although different regimes to partition the orbital space into 1D- and 2D-blocks in an optimal way are observed along the potential energy curves (PEC), a simple strategy for choosing the block types associated to the geminals in each 2D-block is found successful in all cases. This makes the method quite practical to use. Moreover, the optimal geminals can be easily related to classical Lewis structures of chemistry. The quality of the PECs is assessed by calculating equilibrium geometries and fundamental frequencies; the improvement with respect to APSG is discussed.
In our previous study [Kawasaki and Nakatani, Mol. Phys. 123(16), e2449186 (2025)], we proposed the low-rank antisymmetric product of geminals (APG) method, which reconstructs the wavefunction by extracting only the important eigenvalues from the APG wavefunction. However, its practical application was limited by the high computational cost from an orbital optimization process, making higher-rank calculations difficult. In this work, we reformulate the orbital part of the wavefunction using Givens rotation matrices, enabling an analytical treatment of the variational optimization. By combining the low-rank APG with the direct Givens rotation method, we achieved a significant improvement in optimization efficiency. We applied the developed method to small molecular systems and confirmed that it provides high accuracy, while also significantly reducing the computational time compared to conventional methods.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
Abstract The accurate electronic structure calculation for strongly correlated chemical systems requires an adequate description for both static and dynamic electron correlation, and is a persistent challenge for quantum chemistry. In order to account for static and dynamic electron correlations accurately and efficiently, in this work we propose a new method by integrating the density matrix renormalization group (DMRG) method and multi-reference second-order Epstein–Nesbet perturbation theory (ENPT2) with a selected configuration interaction (SCI) approximation. Compared with previous DMRG-based dynamic correlation methods, the DMRG-ENPT2 method extends the range of applicability, allowing us to efficiently calculate systems with very large active space beyond 30 orbitals. We demonstrate this by performing calculations on H 2 S with an active space of (16e, 15o), hexacene with an active space of (26e, 26o) and 2D H 64 square lattice with an active space of (42e, 42o).