Compact Vibrational Wave Functions via Linear Optimization

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r 12, r 13, r 23, r 14, r 24, and r 34. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 12 S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be+, B2+, C3+, ..., Na8+, and Mg9+. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q −1 expansion are derived for the total energies of these three-electron systems.

By combining density-functional theory (DFT) and wave function theory via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions. In particular, we combine here short-range exchange-correlation functionals with a flavor of selected configuration interaction known as configuration interaction using a perturbative selection made iteratively (CIPSI), a scheme that we label RS-DFT-CIPSI. One of the take-home messages of the present study is that RS-DFT-CIPSI trial wave functions yield lower fixed-node energies with more compact multi-determinant expansions than CIPSI, especially for small basis sets. Indeed, as the CIPSI component of RS-DFT-CIPSI is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional CIPSI calculation. Importantly, by performing various numerical experiments, we evidence that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects, hence avoiding the burden of performing a stochastic optimization. Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of μ = 0.5 bohr-1 provides effective error cancellations as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.

Reconciling simplicity and accuracy: compact valence bond wave functions with breathing orbitals

The ground state and first singlet excited state of ethylene, so-called N and V states, respectively, are studied by means of modern valence bond methods. It is found that extremely compact wave functions, made of three VB structures for the N state and four structures for the V state, provide an N → V transition energy of 8.01 eV, in good agreement with experiment (7.88 eV for the N → V transition energy estimated from experiments). Further improvement to 7.96/7.93 eV is achieved at the variational and diffusion Monte Carlo (MC) levels, respectively, VMC/DMC, using a Jastrow factor coupled with the same compact VB wave function. Furthermore, the measure of the spatial extension of the V state wave function, 19.14 a 0 2 , is in the range of accepted values obtained by large-scale state-of-the-art molecular orbital-based methods. The σ response to the fluctuations of the π electrons in the V state, known to be a crucial feature of the V state, is taken into account using the breathing orbital valence bond method, which allows the VB structures to have different sets of orbitals. Further valence bond calculations in a larger space of configurations, involving explicit participation of the σ response, with 9 VB structures for the N state and 14 for the V state, confirm the results of the minimal structure set, yielding an N → V transition energy of 7.97 eV and a spatial extension of 19.16 a 0 2 for the V state. Both types of valence bond calculations show that the V state of ethylene is not fully ionic as usually assumed, but involving also a symmetry-adapted combination of VB structures each with asymmetric covalent π bonds. The latter VB structures have cumulated weights of 18–26 % and stabilize the V state by about 0.9 eV. It is further shown that these latter VB structures, rather than the commonly considered zwitterionic ones, are the ones responsible for the spatial extension of the V state, known to be ca. 50 % larger than the V state.

The ground state and first singlet excited state of ethylene, so-called N and V states, respectively, are studied by means of modern valence bond methods. It is found that extremely compact wave functions, made of three VB structures for the N state and four structures for the V state, provide an N → V transition energy of 8.01 eV, in good agreement with experiment (7.88 eV for the N → V transition energy estimated from experiments). Further improvement to 7.96/7.93 eV is achieved at the variational and diffusion Monte Carlo (MC) levels, respectively, VMC/DMC, using a Jastrow factor coupled with the same compact VB wave function. Furthermore, the measure of the spatial extension of the V state wave function, 19.14 a 0 2 , is in the range of accepted values obtained by large-scale state-of-the-art molecular orbitalbased methods. The σ response to the fluctuations of the π electrons in the V state, known to be a crucial feature of the V state, is taken into account using the breathing orbital valence bond method, which allows the VB structures to have different sets of orbitals. Further valence bond calculations in a larger space of configurations, involving explicit participation of the σ response, with 9 VB structures for the N state and 14 for the V state, confirm the results of the minimal structure set, yielding an N → V transition energy of 7.97 eV and a spatial extension of 19.16 a 0 2 for the V state. Both types of valence bond calculations show that the V state of ethylene is not fully ionic as usually assumed, but involving also a symmetryadapted combination of VB structures each with asymmetric covalent π bonds. The latter VB structures have cumulated weights of 18–26 % and stabilize the V state by about 0.9 eV. It is further shown that these latter VB structures, rather than the commonly considered zwitterionic ones, are the ones responsible for the spatial extension of the V state, known to be ca. 50 % larger than the V state.

The semi-exponential basis set of radial functions (A.M. Frolov, Physics Letters A {\bf 374}, 2361 (2010)) is used for variational computations of bound states in three-electron atomic systems. It appears that semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than it was originally anticipated. In particular, the 40-term Larson's wave function improved with the use of semi-exponential radial basis functions now produces the total energy \linebreak -7.47805413551 $a.u.$ for the ground $1^2S-$state in the ${}^{\infty}$Li atom (only one spin function $\chi_1 = \alpha \beta \alpha - \beta \alpha \alpha$ was used in these calculations). This variational energy is very close to the exact ground state energy of the ${}^{\infty}$Li atom and it substantially lower than the total energy obtained with the original Larson's 40-term wave function (-7.477944869 $a.u.$).

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation---indicated by a larger leading ground-state coefficient---is obtained when combining the symmetric group ${\mathcal{S}}_{n}$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group $\mathrm{U}(n)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

We introduce a local measure of the quality of a trial wave function: the local variance. Using this tool we examine the pair function often adopted to construct wave functions for small helium clusters and apply it to a wave function for 4He2 using the TTY interaction potential. Our analysis shows that this commonly employed functional form should be improved in the short-range region. We introduce a new model based on the short-range behaviour of the exact ground-state solution of a Morse potential. The resulting compact trial wave function for 4He2 recovers 98% of the exact binding energy with only six variational parameters.

The selection of which configurations to include in a configuration interaction (CI) wave function is a compromise between accuracy and computational difficulty. A compact and accurate configuration interaction wave function can be constructed by inclusion of all single and double excitations and certain triple and quadruple excitations chosen in an a priori manner according to how many electrons are placed in several subsets of orbitals. Such a wave function, denoted CISD[TQ], has previously been shown to recover a large fraction of the energy of a CI wave function including all single, double, triple, and quadruple excitations (CISDTQ). A comparison of the molecular geometry and harmonic vibrational frequencies of hydrogen sulfide (H2S) predicted by two CISD[TQ] wave functions and the complete CISDTQ wave function are presented. With the largest basis set used, a triple-ζ plus double polarization basis with an additional set of d-type functions added to hydrogen, and an additional set of f-type functions added to sulfur [TZ2P(f,d)], the CISD[TQ] predictions differ from the CISDTQ by 0.0003 Å in the bond length and by 0.02° in the bond angle. The CISD[TQ] harmonic vibrational frequencies differ by less than 2 cm−1 from the full CISDTQ predictions. These results suggest that the CISD[TQ] wave function is an efficient and accurate truncation of the complete CISDTQ and are particularly impressive considering that with a TZ2P(f,d) basis, the larger CISD[TQ] wave functions included roughly 300 000 configurations while the CISDTQ includes almost nine million.

The performance of a multireference CISD method, CISD[TQ], is compared to that of other approaches which include a large degree of electron correlation, including Brueckner methods. The CISD[TQ] method selects as references all single and double substitutions within an active orbital space. Certain triple and quadruple substitutions from the Hartree–Fock reference are included in the CISD[TQ] wave function as singles and doubles from the selected reference set. This wave function has previously been shown in simpler cases to provide results near to those predicted by the configuration interaction wave function, including all single, double, triple, and quadruple substitutions (CISDTQ). For the challenging multireference case of ozone, the CISD[TQ] wave function yields geometries and harmonic vibrational frequencies with an accuracy similar to the full CCSDT method. These promising results suggest that for difficult multireference problems the CISD[TQ] wave function provides an efficient and accurate approach for approximating the complete CISDTQ.

We derive general bivariational equations of motion (EOMs) for time-dependent wave functions with biorthogonal time-dependent basis sets. The time-dependent basis functions are linearly parameterized and their fully variational time evolution is ensured by solving a set of so-called constraint equations, which we derive for arbitrary wave function expansions. The formalism allows division of the basis set into an active basis and a secondary basis, ensuring a flexible and compact wave function. We show how the EOMs specialize to a few common wave function forms, including coupled cluster and linearly expanded wave functions. It is demonstrated, for the first time, that the propagation of such wave functions is not unconditionally stable when a secondary basis is employed. The main signature of the instability is a strong increase in non-orthogonality, which eventually causes the calculation to fail; specifically, the biorthogonal active bra and ket bases tend toward spanning different spaces. Although formally allowed, this causes severe numerical issues. We identify the source of this problem by reparametrizing the time-dependent basis set through polar decomposition. Subsequent analysis allows us to remove the instability by setting appropriate matrix elements to zero. Although this solution is not fully variational, we find essentially no deviation in terms of autocorrelation functions relative to the variational formulation. We expect that the results presented here will be useful for the formal analysis of bivariational time-dependent wave functions for electronic and nuclear dynamics in general and for the practical implementation of time-dependent CC wave functions in particular.

Electron correlations in molecules can be divided in short range dynamical correlations, long range Van der Waals type interactions, and near degeneracy static correlations. In this work, we analyze for a one-dimensional model of a two-electron system how these three types of correlations can be incorporated in a simple wave function of restricted functional form consisting of an orbital product multiplied by a single correlation function f (r12) depending on the interelectronic distance r12. Since the three types of correlations mentioned lead to different signatures in terms of the natural orbital (NO) amplitudes in two-electron systems, we make an analysis of the wave function in terms of the NO amplitudes for a model system of a diatomic molecule. In our numerical implementation, we fully optimize the orbitals and the correlation function on a spatial grid without restrictions on their functional form. Due to this particular form of the wave function, we can prove that none of the amplitudes vanishes and moreover that it displays a distinct sign pattern and a series of avoided crossings as a function of the bond distance in agreement with the exact solution. This shows that the wave function ansatz correctly incorporates the long range Van der Waals interactions. We further show that the approximate wave function gives an excellent binding curve and is able to describe static correlations. We show that in order to do this the correlation function f (r12) needs to diverge for large r12 at large internuclear distances while for shorter bond distances it increases as a function of r12 to a maximum value after which it decays exponentially. We further give a physical interpretation of this behavior.

Traditionally, because of the limit of full configuration interaction, complete active space (CAS) theory is most often used to model bond dissociation and other dynamical processes where the multi-reference character becomes important. Inconveniently, the CAS method is highly dependent on the choice of active space and, therefore, inherently non-black-box, in addition to the exponential scaling with respect to electrons and orbitals. This illustrates the need for methods that can accurately treat multi-reference electronic structure problems without significant dependence on input parameters. Selected configuration interaction (SCI) methods have experienced a revival in recent years because of their independence of these predicaments. SCI methods aim to exploit the sparsity of the full configuration interaction space to identify all relevant electronic configurations and, therefore, keep the wave function as compact as possible while still representing the total multi-reference electronic structure accurately. In this work, we take the recent achievement by Gao et al. to run full configuration interaction on the propane molecule in a minimal basis set (23 electrons in 26 orbitals) as an occasion to demonstrate that our SCI methods implemented in the GeneralSCI program package can achieve high energetic accuracy in conjunction with very compact wave functions, which considerably alleviate computational cost. Furthermore, we show the good performance of our SCI methods in reproducing a propane bond dissociation surface and energy. This illustrates that SCI methods can be readily applied to problems in chemical reactivity.

The method of the preceding paper is used to construct compact and accurate integral-transform wave functions for the four lowest excited states of the helium-like ions. Convergence studies of the energy and various expectation values are presented for the 2 /sup 1/S and 2 /sup 3/S states of the helium atom. Parallelotope parameters and energies are given for the 2 /sup 1/S, 2 /sup 3/S, 2 /sup 1/P, and 2 /sup 3/P states of the ions from He through Mg/sup 10 +/. A brief discussion of the prospects for extending the method to many-electron systems is included. (AIP)

Compact and accurate wave functions can be constructed by quantum Monte Carlo methods. Typically, these wave functions consist of a sum of a small number of Slater determinants multiplied by a Jastrow factor. In this paper we study the importance of including high-order, nucleus-three-electron correlations in the Jastrow factor. An efficient algorithm based on the theory of invariants is used to compute the high-body correlations. We observe significant improvements in the variational Monte Carlo energy and in the fluctuations of the local energies but not in the fixed-node diffusion Monte Carlo energies. Improvements for the ground states of physical, fermionic atoms are found to be smaller than those for the ground states of fictitious, bosonic atoms, indicating that errors in the nodal surfaces of the fermionic wave functions are a limiting factor.