First Ionization Energy as the Asymptotic Limit of the Average Local Electron Energy.

The first vertical ionization energy of an atom or molecule is encoded in the rate of exponential decay of the exact natural orbitals. For natural orbitals represented in terms of Gaussian basis functions, this property does not hold even approximately. We show that it is nevertheless possible to deduce the first ionization energy from the long-range behavior of Gaussian-basis-set wave functions by evaluating the asymptotic limit of a quantity called the average local electron energy (ALEE), provided that the most diffuse functions of the basis set have a suitable shape and location. The ALEE method exposes subtle qualitative differences between seemingly analogous Gaussian basis sets and complements the extended Koopmans theorem by being robust in situations where the one-electron reduced density matrix is ill-conditioned.

What Is the Accuracy Limit of Adiabatic Linear-Response TDDFT Using Exact Exchange-Correlation Potentials and Approximate Kernels?

Calculation of vertical excitation energies by the adiabatic linear-response time-dependent density-functional theory (TDDFT) requires static Kohn-Sham potentials and exchange-correlation kernels. When these quantities are derived from standard density-functional approximations (DFA), mean absolute errors (MAE) of the method are known to range from 0.2 eV to over 1 eV, depending on the functional and type of excitation. We investigate how the performance of TDDFT varies when increasingly accurate exchange-correlation potentials derived from Hartree-Fock (HF) and post-HF wavefunctions are combined with different approximate kernels. The lowest MAEs obtained in this manner for valence excitations are about 0.15-0.2 eV, which appears to be the practical limit of the accuracy of TDDFT that can be achieved by improving the Kohn-Sham potentials alone. These findings are consistent with previous reports on the benefits of accurate exchange-correlation potentials in TDDFT, but provide new insights and afford more definitive conclusions.

Visualizing atomic sizes and molecular shapes with the classical turning surface of the Kohn-Sham potential.

The Kohn-Sham potential [Formula: see text] is the effective multiplicative operator in a noninteracting Schrödinger equation that reproduces the ground-state density of a real (interacting) system. The sizes and shapes of atoms, molecules, and solids can be defined in terms of Kohn-Sham potentials in a nonarbitrary way that accords with chemical intuition and can be implemented efficiently, permitting a natural pictorial representation for chemistry and condensed-matter physics. Let [Formula: see text] be the maximum occupied orbital energy of the noninteracting electrons. Then the equation [Formula: see text] defines the surface at which classical electrons with energy [Formula: see text] would be turned back and thus determines the surface of any electronic object. Atomic and ionic radii defined in this manner agree well with empirical estimates, show regular chemical trends, and allow one to identify the type of chemical bonding between two given atoms by comparing the actual internuclear distance to the sum of atomic radii. The molecular surfaces can be fused (for a covalent bond), seamed (ionic bond), necked (hydrogen bond), or divided (van der Waals bond). This contribution extends the pioneering work of Z.-Z. Yang et al. [Yang ZZ, Davidson ER (1997) Int J Quantum Chem 62:47-53; Zhao DX, et al. (2018) Mol Phys 116:969-977] by our consideration of the Kohn-Sham potential, protomolecules, doubly negative atomic ions, a bond-type parameter, seamed and necked molecular surfaces, and a more extensive table of atomic and ionic radii that are fully consistent with expected periodic trends.

Construction of Fermi Potentials from Electronic Wave Functions.

The Fermi potential, vF(r), is the nonclassical part of the multiplicative effective potential appearing in the one-particle Schrödinger-type equation for the square root of the electron density. The usual way of constructing vF(r) by inverting that equation produces unsatisfactory results when applied to electron densities expanded in Gaussian basis sets. We suggest a different method that is based on an explicit formula for vF(r) in terms of the interacting one- and two-electron reduced density matrices of the system. This method is exact in the basis-set limit and yields accurate approximations to the basis-set-limit vF(r) when applied to reduced density matrices represented in terms of finite basis sets. Illustrative applications involve atomic and molecular wave functions generated at various levels of ab initio theory. It is also shown how to construct the Pauli and exchange-correlation potentials of any system starting with only vF(r).

Exact exchange-correlation potentials of singlet two-electron systems.

We suggest a non-iterative analytic method for constructing the exchange-correlation potential, vXC(r), of any singlet ground-state two-electron system. The method is based on a convenient formula for vXC(r) in terms of quantities determined only by the system's electronic wave function, exact or approximate, and is essentially different from the Kohn-Sham inversion technique. When applied to Gaussian-basis-set wave functions, the method yields finite-basis-set approximations to the corresponding basis-set-limit vXC(r), whereas the Kohn-Sham inversion produces physically inappropriate (oscillatory and divergent) potentials. The effectiveness of the procedure is demonstrated by computing accurate exchange-correlation potentials of several two-electron systems (helium isoelectronic series, H2, H3+) using common ab initio methods and Gaussian basis sets.

Improved method for generating exchange-correlation potentials from electronic wave functions.

Ryabinkin, Kohut, and Staroverov (RKS) [Phys. Rev. Lett. 115, 083001 (2015)] devised an iterative method for reducing many-electron wave functions to Kohn-Sham exchange-correlation potentials, vXC(𝐫). For a given type of wave function, the RKS method is exact (Kohn-Sham-compliant) in the basis-set limit; in a finite basis set, it produces an approximation to the corresponding basis-set-limit vXC(𝐫). The original RKS procedure works very well for large basis sets but sometimes fails for commonly used (small and medium) sets. We derive a modification of the method's working equation that makes the RKS procedure robust for all Gaussian basis sets and increases the accuracy of the resulting exchange-correlation potentials with respect to the basis-set limit.

Note: A pure-sampling quantum Monte Carlo algorithm with independent Metropolis.

Recently, Ospadov and Rothstein published a pure-sampling quantum Monte Carlo algorithm (PSQMC) that features an auxiliary Path Z that connects the midpoints of the current and proposed Paths X and Y, respectively. When sufficiently long, Path Z provides statistical independence of Paths X and Y. Under those conditions, the Metropolis decision used in PSQMC is done without any approximation, i.e., not requiring microscopic reversibility and without having to introduce any G(x → x'; τ) factors into its decision function. This is a unique feature that contrasts with all competing reptation algorithms in the literature. An example illustrates that dependence of Paths X and Y has adverse consequences for pure sampling.

A pure-sampling quantum Monte Carlo algorithm.

The objective of pure-sampling quantum Monte Carlo is to calculate physical properties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the exact wave function. To achieve this objective, we report a pure-sampling algorithm that combines features of forward walking methods of pure-sampling and reptation quantum Monte Carlo (RQMC). The new algorithm accurately samples properties from the mixed and pure distributions simultaneously in runs performed at a single set of time-steps, over which extrapolation to zero time-step is performed. In a detailed comparison, we found RQMC to be less efficient. It requires different sets of time-steps to accurately determine the energy and other properties, such as the dipole moment. We implement our algorithm by systematically increasing an algorithmic parameter until the properties converge to statistically equivalent values. As a proof in principle, we calculated the fixed-node energy, static α polarizability, and other one-electron expectation values for the ground-states of LiH and water molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We found excellent agreement with the accepted values for the energy and a variety of other properties for those systems.

Ground-state properties of LiH by reptation quantum Monte Carlo methods.

We apply reptation quantum Monte Carlo to calculate one- and two-electron properties for ground-state LiH, including all tensor components for static polarizabilities and hyperpolarizabilities to fourth-order in the field. The importance sampling is performed with a large (QZ4P) STO basis set single determinant, directly obtained from commercial software, without incurring the overhead of optimizing many-parameter Jastrow-type functions of the inter-electronic and internuclear distances. We present formulas for the electrical response properties free from the finite-field approximation, which can be problematic for the purposes of stochastic estimation. The α, γ, A and C polarizability values are reasonably consistent with recent determinations reported in the literature, where they exist. A sum rule is obeyed for components of the B tensor, but B(zz,zz) as well as ß(zzz) differ from what was reported in the literature.