For nonorthogonal electron group functions, the arrow diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators . The total wavefunction of the system is represented as an antisymmetrized product of nonorthogonal many-electron group functions Φ I of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the diagram expansion be divided by the normalization integral which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree–Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).
For nonorthogonal electron group functions, the arrow diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators . The total wavefunction of the system is represented as an antisymmetrized product of nonorthogonal many-electron group functions Φ I of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the diagram expansion be divided by the normalization integral which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree–Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).
Arrow diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $$, of symmetrical operators, O^, in quantum chemistry when the total system wavefunction Ψ is represented as an antisymmetrised product of overlapping many-electron group functions, ΦA, corresponding to each part (group) A of the system: Ψ=A^∏AΦA. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the diagram expansion is to be divided by the normalisation integral S=, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.
Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.
Wavefunctions for large interacting electron systems lose their meaning due to an exponential growth of the dimensions of Hilbert space with increasing electron number. In order to base electronic-structure calculations for solids on wavefunctions instead on density functional theory one has to resolve this exponential wall problem (EWP). It is shown that the origin of it is the multiplicative character of a wavefunction with respect to independent systems A and B, i.e., ψA/B = ψA ⨂ ψB. The EWP is avoided, if we characterize the system by an additive quantity like an action instead. This can be done by describing the system through the fluctuations of a mean-field state imposed by the interactions. The operators defining these fluctuations span an operator- or Liouville space. In order to obtain additive quantities the metric in Liouville space must be a special one, i.e., based on cumulants. The situation resembles the one of a classical gas, where the interaction contributions to the energy are included by a linked cluster expansion. In this way the EWP is avoided and one obtains a solid basis for wavefunction-based electronic-structure calculations of large systems. Examples are given for the application of the theory.
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
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.