Finite Orbital Basis Research Articles

Assessment of multireference approaches to explicitly correlated full configuration interaction quantum Monte Carlo.

The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method has proved able to provide near-exact solutions to the electronic Schrödinger equation within a finite orbital basis set, without relying on an expansion about a reference state. However, a drawback to the approach is that being based on an expansion of Slater determinants, the FCIQMC method suffers from a basis set incompleteness error that decays very slowly with the size of the employed single particle basis. The FCIQMC results obtained in a small basis set can be improved significantly with explicitly correlated techniques. Here, we present a study that assesses and compares two contrasting "universal" explicitly correlated approaches that fit into the FCIQMC framework: the [2]R12 method of Kong and Valeev [J. Chem. Phys. 135, 214105 (2011)] and the explicitly correlated canonical transcorrelation approach of Yanai and Shiozaki [J. Chem. Phys. 136, 084107 (2012)]. The former is an a posteriori internally contracted perturbative approach, while the latter transforms the Hamiltonian prior to the FCIQMC simulation. These comparisons are made across the 55 molecules of the G1 standard set. We found that both methods consistently reduce the basis set incompleteness, for accurate atomization energies in small basis sets, reducing the error from 28 mEh to 3-4 mEh. While many of the conclusions hold in general for any combination of multireference approaches with these methodologies, we also consider FCIQMC-specific advantages of each approach.

Reply to “Comment on ‘Nonanalyticity of the optimized effective potential with finite basis sets’ ”

The Comment by Friedrich et al. does not dispute the central result of our paper [Phys. Rev. A 85, 052508 (2012)] that nonanalytic behavior is present in long-established mathematical pathologies arising in the solution of finite basis optimized effective potential (OEP) equations. In the Comment, the terms ``balancing of basis sets'' and ``basis-set convergence'' imply a particular order towards the limit of a large orbital basis sets where the large-orbital-base limit is always taken first, before the large-auxiliary-base limit, until overall convergence is achieved, at a high computational cost. The authors claim that, on physical grounds, this order of limits is not only sufficient, but also necessary in order to avoid the mathematical pathologies. In response to the Comment, we remark that it is already written in our paper that the nonanalyticity trivially disappears with large orbital basis sets. We point out that the authors of the Comment give an incorrect proof of this statement. We also show that the order of limits towards convergence of the potential is immaterial. A recent paper by the authors of the Comment proposes a partial correction for the incomplete orbital basis error in the full-potential linearized augmented-plane-wave method. Similar to the correction developed in our paper, this correction also benefits from an effectively complete orbital basis, even though only a finite orbital basis is employed in the calculation. This shows that it is unnecessary to take, in practice, the limit of an infinite orbital basis in order to avoid mathematical pathologies in the OEP. Our paper is a significant contribution in that direction with general applicability to any choice of basis sets. Finally, contrary to an allusion in the abstract and assertions in the main text of the Comment that unphysical oscillations of the OEP are supposedly attributed to the common energy denominator approximation, in fact, such anomalies arise with the full treatment of the small eigenvalues of the density response function. This characteristic of the finite basis OEP is well known in the literature but also is clearly demonstrated in our paper.

Optimized unrestricted Kohn–Sham potentials from ab initio spin densities

The reconstruction of the exchange–correlation potential from accurate ab initio electron densities can provide insights into the limitations of the currently available approximate functionals and provide guidance for devising improved approximations for density-functional theory (DFT). For open-shell systems, the spin density is introduced as an additional fundamental variable in spin-DFT. Here, we consider the reconstruction of the corresponding unrestricted Kohn–Sham (KS) potentials from accurate ab initio spin densities. In particular, we investigate whether it is possible to reconstruct the spin exchange–correlation potential, which determines the spin density in unrestricted KS-DFT, despite the numerical difficulties inherent to the optimization of potentials with finite orbital basis sets. We find that the recently developed scheme for unambiguously singling out an optimal optimized potential [Ch. R. Jacob, J. Chem. Phys. 135, 244102 (2011)10.1063/1.3670414] can provide such spin potentials accurately. This is demonstrated for two test cases, the lithium atom and the dioxygen molecule, and target (spin) densities from full configuration interaction and complete active space self-consistent field calculations, respectively.

Optimized effective potential method: Is it possible to obtain an accurate representation of the response function for finite orbital basis sets?

The optimized effective potential (OEP) equations are solved in a matrix representation using the orbital products of occupied and virtual orbitals for the representation of both the local potential and the response function. This results in a direct relationship between the matrix elements of local and nonlocal operators for the exchange-correlation potential. The effect of the truncation of the number of such products in the case of finite orbital basis sets on the OEP orbital and total energies and on the spectrum of eigenvalues of the response function is examined. Test calculations for Ar and Ne show that rather large AO basis sets are needed to obtain an accurate representation of the response function.

Variational collapse of the optimized effective potential method with an orbital-dependent exchange-correlation functional based on second order perturbation theory

It is well known, that second-order perturbation theory (PT2) breaks down when the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied MO (LUMO) becomes too low. It is demonstrated that when the local Kohn–Sham (KS) potential v s is approximated by an expansion in a set of basis functions, exact HOMO–LUMO degeneracy occurs, if a finite orbital basis set is used. Numerical evidence is given for the He atom, which until now stood out as a ‘safe’ simple case. Variational collapse of the optimized effective potential (OEP) method with the PT2 functional in a finite orbital basis may be expected to be a common phenomenon. We also show variational breakdown with a non-perturbative functional when the HOMO–LUMO gap is used to regulate the contribution of virtual orbitals.

Kohn—Sham density-functional theory within a finite basis set

The Kohn—Sham self-consistent equations, using a finite orbital basis expansion, are formulated for exchange-correlation functionals which depend on local densities and their gradients. It is shown that these can be solved iteratively without evaluation of density Hessians. A general expansion is given for the energy gradient (with respect to nuclear motion) after self-consistency has been achieved.

Some comments on the electrostatic potential of a molecule

In this article we discuss several principles and tools which should expedite description of the electrostatic potentials and electrostatic interactions of molecules, and show that these also lead to some rather remarkable results in the theory of the irreducible representations of the full rotation group SO(3). First, by representing a molecule's charge-density matrix over a basis of atomic-like orbitals (on the various atoms), we observe that outside its charge distribution the molecule's electrostatic potential is exactly the same as if that charge distribution were merely a sum (and in the case of a finite orbital basis, this is a finite sum) of point multipoles on each of the atomic centers and line multipoles on the line segments joining each of those atomic centers. Possible methods of approximating the field of these line charges and line multipoles, as if they were due to point charges and point multipoles, are discussed. The calculation of the interaction of point multipoles of high order, as is necessary for this procedure to successfully calculate the interaction of arbitrarily oriented molecules, motivates our second topic. Here we present a differential operator which, when acting on the 3-dimensional delta function, produces the source density for a scalar field that is exactly an (l,m) multipole field. Using the Hermitian adjoint of this operator, we express the interaction of this (l,m) multipole with an external scalar field as the result of this differential operator acting on that external field at the location of this multipole source. Irreducible representation matrices of the full rotation group are then used, together with these relations, to simplify the interaction of two arbitrarily oriented multipoles of any orders. Finally, we use the representation of the Condon and Shortley “raising and lowering” relations on eigenstates of the z-component of angular momentum, in an orientation that is not aligned with its fundamental basis states, to generate recursion relations that allow simple calculations of the irreducible representation matrices of the full rotation group, SO(3), and the special unitary group, SU(2). From these recursion relations we display some useful symmetry properties of our parameterization of these matrices, that allow the entire matrix to be very simply generated from an explicit calculation of only about 1/8 of its elements. © 1992 John Wiley & Sons, Inc.

Derivation of an extended geminal model

An extended geminal model is derived by applying perturbation theory to a matrix representation of the electronic Hamiltonian. The matrix representation is formed by a complete set of orthogonal configuration functions generated from a given finite orbital basis set. By construction, the function defined by the APSG approximation (antisymmetric product of strongly orthogonal geminals) is an element of this set of configuration functions. Considering the nondiagonal part of the Hamiltonian matrix as the perturbation, our model is obtained by a partitioning and reordering of the infinite order Rayleigh–Schrödinger perturbation expansion. The model has a hierarchical structure, is size extensive, and has favorable properties with respect to interpretation. In a test calculation on the water molecule, it is demonstrated that the extended geminal model truncated at the double pair correction level, recovers 100.8% of the correlation energy defined by the corresponding full CI calculation.

Time-dependent multiconfigurational Hartree–Fock theory

A time dependent version of multiconfigurational Hartree–Fock theory is formulated as an extension of the time-dependent Hartree–Fock approximation. The time variation of the reference state is calculated using a form of Frenkel’s variation principle, which was suggested by Langhoff, Epstein, and Karplus. A linearized form of the equations provides linear response functions, which transform correctly under contact transformations generated by one particle operators. A consequence of this feature is that the various formally equivalent expressions for the oscillator strengths of electronic transitions will give the same result, apart from truncation errors when a finite orbital basis is used. Example calculations are carried out for some π electron systems within the Pariser–Parr–Pople model using a correlated reference state suggested by Clementi and Veillard.

Density matrix methods in orbital optimization for MCSCF calculations

AbstractThe problem considered is that of selecting the finite orbital basis which will minimize the energy in a given size CI calculation. (1) A one‐body operator is defined which has as eigenfunctions the desired optimal basis. The operator is defined in terms of the basis which leads to a self‐consistency problem of Hartree‐Fock type. (2) A method of successive orbital rotations is defined which is shown to have desirable convergence properties.