Yves A. Bernard

General formulation of spin-flip time-dependent density functional theory using non-collinear kernels: Theory, implementation, and benchmarks

We report an implementation of the spin-flip (SF) variant of time-dependent density functional theory (TD-DFT) within the Tamm-Dancoff approximation and non-collinear (NC) formalism for local, generalized gradient approximation, hybrid, and range-separated functionals. The performance of different functionals is evaluated by extensive benchmark calculations of energy gaps in a variety of diradicals and open-shell atoms. The benchmark set consists of 41 energy gaps. A consistently good performance is observed for the Perdew-Burke-Ernzerhof (PBE) family, in particular PBE0 and PBE50, which yield mean average deviations of 0.126 and 0.090 eV, respectively. In most cases, the performance of original (collinear) SF-TDDFT with 50-50 functional is also satisfactory (as compared to non-collinear variants), except for the same-center diradicals where both collinear and non-collinear SF variants that use LYP or B97 exhibit large errors. The accuracy of NC-SF-TDDFT and collinear SF-TDDFT with 50-50 and BHHLYP is very similar. Using PBE50 within collinear formalism does not improve the accuracy.

Intracule functional models. Part III. The dot intracule and its Fourier transform.

The dot intracule D(x) of a system gives the Wigner quasi-probability of finding two of its electrons with u.v = x, where u and v are the interelectronic distance vectors in position and momentum space, respectively. In this paper, we discuss D(x) and show that its Fourier transform d(k) can be obtained in closed form for any system whose wavefunction is expanded in a Gaussian basis set. We then invoke Parsevals theorem to transform our intracule-based correlation energy method into a d(k)-based model that requires, at most, a one-dimensional quadrature.

Distribution of r12 · p12 in quantum systems

We introduce the two-particle probability density X(x) of x= r 12· p 12=( r 1 − r 2)·( p 1 − p 2). The fundamental equations involved in the derivation of this new intracule X(x), which we call the Posmom intracule, are derived and we show how to derive X(x) from the many-particle wave-function. We contrast it with the Dot intracule [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, Phys. Chem. Chem. Phys. 10, 3447 (2008)] which can be derived from the Wigner distribution and show the relationships between the Posmom intracule and the one-particle Posmom density [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, J. Phys. Chem. A 114, 11984 (2010)]. To illustrate the information provided by the Posmom intracule, we apply this new formalism to various two-electron systems: the three-dimensional parabolic quantum dot, the helium-like ions and the ground and excited states of the helium atom.

Distribution of r·p in atomic systems.

We present formulas for computing the probability distribution of the posmom s = r · p in atoms, when the electronic wave function is expanded in a single particle Gaussian basis. We study the posmom density, S(s), for the electrons in the ground states of 36 lightest atoms (H-Kr) and construct an empirical model for the contribution of each atomic orbital to the total S(s). The posmom density provides unique insight into types of trajectories electrons may follow, complementing existing spectroscopic techniques that provide information about where electrons are (X-ray crystallography) or where they go (Compton spectroscopy). These, a priori, predictions of the quantum mechanically observable posmom density provide an challenging target for future experimental work.

Compact expressions for spherically averaged position and momentum densities.

Compact expressions for spherically averaged position and momentum density integrals are given in terms of spherical Bessel functions (j(n)) and modified spherical Bessel functions (i(n)), respectively. All integrals required for ab initio calculations involving s, p, d, and f-type Gaussian functions are tabulated, highlighting a neat isomorphism between position and momentum space formulae. Spherically averaged position and momentum densities are calculated for a set of molecules comprising the ten-electron isoelectronic series (Ne-CH(4)) and the eighteen-electron series (Ar-SiH(4), F(2)-C(2)H(6)).

Distribution of

We introduce the two-particle probability density X(x) of x= r 12· p 12=( r 1 − r 2)·( p 1 − p 2). The fundamental equations involved in the derivation of this new intracule X(x), which we call the Posmom intracule, are derived and we show how to derive X(x) from the many-particle wave-function. We contrast it with the Dot intracule [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, Phys. Chem. Chem. Phys. 10, 3447 (2008)] which can be derived from the Wigner distribution and show the relationships between the Posmom intracule and the one-particle Posmom density [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, J. Phys. Chem. A 114, 11984 (2010)]. To illustrate the information provided by the Posmom intracule, we apply this new formalism to various two-electron systems: the three-dimensional parabolic quantum dot, the helium-like ions and the ground and excited states of the helium atom.

r_{12} \cdot p_{12}

We introduce the two-particle probability density X(x) of x= r 12· p 12=( r 1 − r 2)·( p 1 − p 2). The fundamental equations involved in the derivation of this new intracule X(x), which we call the Posmom intracule, are derived and we show how to derive X(x) from the many-particle wave-function. We contrast it with the Dot intracule [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, Phys. Chem. Chem. Phys. 10, 3447 (2008)] which can be derived from the Wigner distribution and show the relationships between the Posmom intracule and the one-particle Posmom density [Y.A. Bernard, D.L. Crittenden, and P.M.W. Gill, J. Phys. Chem. A 114, 11984 (2010)]. To illustrate the information provided by the Posmom intracule, we apply this new formalism to various two-electron systems: the three-dimensional parabolic quantum dot, the helium-like ions and the ground and excited states of the helium atom.