物理化学学报  2018, Vol. 34 Issue (5): 528-536    DOI: 10.3866/PKU.WHXB201710111 所属专题： 密度泛函理论中的化学概念特刊
 论文
Chemical Bonding and Interpretation of Time-Dependent Electronic Processes with Maximum Probability Domains
SAVIN Andreas*()
Chemical Bonding and Interpretation of Time-Dependent Electronic Processes with Maximum Probability Domains
Andreas SAVIN*()
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Tools have been designed obtain information about chemical bonds from quantum mechanical calculations. They work well for solutions of the stationary Schrödinger equation, but it is not clear whether Lewis electron pairs they aim to reproduce survive in time-dependent processes, in spite of the underlying Pauli principle being obeyed in this regime. A simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, is used to study this problem, because it allows presenting the detailed structure of the wave function. It is shown that ⅰ) oscillations persisting after the Hamiltonian stopped changing produce for certain time intervals states where Lewis electron pairs are spatially separated, and ⅱ) methods (like density analysis, or the electron localization function) that are widely used for describing bonding in the stationary case, have limitations in such situations. An exception is provided by the maximum probability domain (the spatial domain that maximizes the probability to find a given number of particles in it). It is conceptually simple, and satisfactorily describes the phenomenon.

Abstract:

Tools have been designed obtain information about chemical bonds from quantum mechanical calculations. They work well for solutions of the stationary Schrödinger equation, but it is not clear whether Lewis electron pairs they aim to reproduce survive in time-dependent processes, in spite of the underlying Pauli principle being obeyed in this regime. A simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, is used to study this problem, because it allows presenting the detailed structure of the wave function. It is shown that ⅰ) oscillations persisting after the Hamiltonian stopped changing produce for certain time intervals states where Lewis electron pairs are spatially separated, and ⅱ) methods (like density analysis, or the electron localization function) that are widely used for describing bonding in the stationary case, have limitations in such situations. An exception is provided by the maximum probability domain (the spatial domain that maximizes the probability to find a given number of particles in it). It is conceptually simple, and satisfactorily describes the phenomenon.

Key words: Chemical bond    Time-dependent Schrödinger equation    Particle in a box with opaque wall

 Fig 1  One-particle eigenfunctions of the stationary Schrödinger equation for a particle in a box with an opaque wall; symmetric solutions $u_+$, for $n=1$ (top), for $n=2$ (center), for different values of the opacity parameter $a$, and antisymmetric solutions $u_-$ (bottom), for $n=1, 2$. Fig 2  Slow change of the Hamiltonian with time. From top to bottom: a) absolute value of the wave function squared, as function of the coordinates of the particles, $x_1, x_2$, b) localized molecular orbitals, c) density, d) minus the second derivative of the density, e) the electron localization function, f) the probability to find one, and only one electron between $x_<$ and $x_>$; left: impenetrable wall ($a=\infty$), right: wall has vanished ($a=0$). Fig 3  Absolute value of the wave function squared, for a sudden change of the Hamiltonian as a function of the coordinates of the particles, $x_1, x_2$. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 4  Absolute value of the square of the localized orbitals, for a sudden change of the Hamiltonian. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 5  Density, $\rho$, for a sudden change of the Hamiltonian. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 6  $-\partial_x^2 \rho$, for a sudden change of the Hamiltonian. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 7  Electron localization function, $\eta$, for a sudden change of the Hamiltonian. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 8  Probability to find one, and only one electron between $x_<$ and $x_>$, for a sudden change of the Hamiltonian. The time after the change of the Hamiltonian is given in atomic units (1 a.u. $\approx 24$ attoseconds), for $L=1$. Fig 9  Time-dependent orbitals squared, reduced to the $n=1, 2$ components, at times $t=0, T, \ldots$ (upper panels), and $t=T/2, 3 T/2, \ldots$ (lower panels); left for symmetry-adapted orbitals (full lines: $u_+$, dashed lines: $u_-$), right for localized orbitals; same scale in all panels. Fig 10  Probability to find one, and only one electron in a half-box (between $x_<=0$ and $x_>=L$), dashed curve, and that for $x_<=0.3 L$ and $x_>=0.3 L$ (full curve), as a function of time after the Hamiltonian stopped changing (given in atomic units, 1 a.u. $\approx 24$ attoseconds; for $L=10$, and the the wall made transparent in $\tau\approx10$ femtoseconds). Fig 11  Eigenvalues of the stationary Schrödinger equation for a particle in a box with a wall having an opacity increasing with $a$; for $n=1, 2$; those corresponding to the symmetric eigenfunctions $u_+$ are shown with full lines; those of the antisymmetric eigenfunctions $u_-$ do not depend on $a$ and are shown as horizontal dashed lines.