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Chemical Bonding and Interpretation of Time-Dependent Electronic Processes with Maximum Probability Domains

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Chemical Bonding and Interpretation of Time-Dependent Electronic Processes with Maximum Probability Domains

**收稿日期:** 2017-09-1
**接受日期:** 2017-09-25

**Corresponding authors:**

**Received:** 2017-09-1
**Accepted:** 2017-09-25

**摘要**

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.

**关键词：**

**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.

**Keywords：**

**本文引用格式**

SAVIN Andreas. *物理化学学报*[J], 2018, 34(5): 528-536 doi:10.3866/PKU.WHXB201710111

SAVIN Andreas. * Acta Physico-Chimica Sinica*[J], 2018, 34(5): 528-536 doi:10.3866/PKU.WHXB201710111

## 1 Introduction

Many tools have been developed to describe chemical bonding using quantum mechanics. But chemical bonding changes during structural modifications of the molecules. Does assigning spatial domains to electron pairs (the Lewis perspective) survive in time-dependent processes? Usual chemical routine uses curved arrows, suggesting that this is the case. Quantum chemical calculations performed along the reaction path tend to confirm it. But is this adiabatic picture correct?

This paper uses a simple model, of two independent particles of the same spin, in a one-dimensional box.(As the formation of the Lewis pairs is mainly due to the Pauli principle, and only independent particles are discussed in this paper, the treatment of two electrons of the same spin is easily transposed to the treatment of two electron pairs.) At start, each of the the particles is confined to a half-box. The wall between boxes becomes transparent with time, allowing the particle to pass from one half-box to the other. After some time,

Using a "reasonable" definition, one can attribute a spatial domain to one of the electrons, the other one being in the remaining space available. This evidently works when the wall is impenetrable. One may naively believe that making the separation wall vanish does not qualitatively change the situation, that the Pauli principle forces the two electron pairs to remain as such, whether they are separated by a wall, or not. However, as we consider a model for a chemical reaction, we should look at the influence of time on the electron localization domain, and whether it affects our perception of electron localization.

The time evolution is computed using

(1) the adiabatic approximation, valid when the Hamiltonian changes very slowly with time,

(2) the sudden approximation, valid when the change of the Hamiltonian is fast,

(3) an explicit solution of the time-dependent Schrödinger equation, for a finite basis set, and given parameters of the system.

For a more precise definition of "slow" and "fast", see, e.g. ^{1}, Section ⅩⅦ.

The calculations below show that with the last two approaches, for certain time intervals, electrons are not essentially confined to the half-boxes, in contrast to a Lewis-like concept. One can find that one electron (or electron pair) is located in the center of the box, while the other is delocalized over the remaining left and right parts.

The simplicity of the model allows presenting the detailed structure of the wave function. Pictures are presented using other interpretative tools that can also be used for more complicated systems (localized orbitals, the density, and its second derivative, the electron localization function, and the maximum probability domains). It is concluded that the latter method is preferable to describe time-dependent processes, although one should keep in mind that the present calculations are far from being representative for real systems.

### 2 System

### 2.1 Hamiltonian

A one-dimensional box stretching from ^{2}, problems 19, 20). The potential is given by:

and is infinite outside this interval. The parameter

### 2.2 Stationary solutions

The solution of the stationary Schrödinger equation for this potential is analytically known. By the symmetry of the potential

The antisymmetric solutions have a node at

### Fig 1

**Fig 1
One-particle eigenfunctions of the stationary Schrödinger equation for a particle in a box with an opaque wall; symmetric solutions **

### 2.3 A triplet non-interacting two particle system

The model system studied in this paper consists of two non-interacting fermions, in a triplet state. In fact, it stands for a system for two non-interacting electron pairs in a singlet state. Having another two electrons with opposite spin changes little to the problem, as the anti-symmetrization needs to be done only among particles of the same spin. The properties of this non-interacting system can be computed from a wave function that is a product of two identical two-by-two Slater determinants, one for each spin (see, e.g. ^{3}). It is thus sufficient to analyze only one of them, the properties of the four-electron system being understood easily from those of the same-spin two electron system. For example, if we have the density of the system with two spin-up electrons in the triplet state, we just have to multiply it by two to obtain that of the four-electron system.

The repulsion between electrons has been neglected because the formation of electron pairs is not due to electron repulsion. The intuition of Lewis was that Coulomb's law is not valid at short distances, and that "each pair of electrons has a tendency to be drawn together" ^{4}. Although the explanation given by Lewis is not correct, such an effect is seen in mean-field models like Hartree-Fock; localized orbitals with different spin are pairwise identical in the spatial part. It is the Pauli principle that keeps the electrons with same spin apart, and it acts whether or not they interact. Opposite spin electrons can share the same spatial domain, and can form the pairs described by Lewis. In fact, many of the tools used to analyze the chemical bond only exploit the Pauli principle.

Another reason not to introduce repulsion in the present calculations is that there is not a clear way how repulsion should be treated in one dimension. The Coulomb interaction in one dimension, ^{5}). Physically, this is easy to understand: electrons can better avoid each other in three dimensions than in one dimension.

### 2.4 Analogies

In order to see a connection to chemistry, we can imagine some analogue. For example, one could consider two He atoms getting closer. From the Lewis pairing perspective, nothing interesting can be expected: even for He

In analogy to a molecule formation in time, we start with the particles separated by an infinite wall,

### 3 Tools to analyze the electron distribution

There are many tools to analyze the electronic structure. Just a few are used below, and are now shortly described.

### 3.1 Wave function

One can analyze the wave function. In general, it has a too high dimension. For our example, it is only in two dimensions (the coordinate of each of the particles), and can be easily plotted.

In order to avoid the dimensionality problem, Artmann ^{6} proposed to locate the maxima of the wave function. This is a very appealing proposal, well adapted to method like Quantum Monte Carlo ^{7}. It has the disadvantage that the wave function can present several maxima, and one has to choose among them. This can be avoided in many practical situations by choosing a domain around them ^{8}.

### 3.2 Maximum probability domains

One way to define a spatial domain is to consider the one that maximizes the probability to have a given number of particles, ^{9}, the "maximum probability domains" (MPDs). In our example, we search for a domain

is maximal.

### 3.3 Density

A simple three-dimensional quantity is the electron density,

Its analysis and use has been much promoted by Bader ^{10}. The particle density should not be confused with a probability density, as

This integral over the density gives the average number of particles in

### 3.4 Second density derivative

The maxima of ^{10}, Section 7.1.4). Here, as our system is in one dimension,

### 3.5 Electron localization function

Another popular quantity to detect the Lewis pairs is the electron localization function (ELF) ^{11}. It is a function defined in each point of space, takig values between 0 and 1. For regions where electrons localize, the values of ELF should be large. It has been generalized to time-dependent processes, TDELF ^{12}. In this paper, we use a formula that is modified for particles in one dimension. The explicit expression of ELF is given in the Appendix B.

As we deal with independent particles, we do not have to worry about generalizations of ELF for wave functions beyond a single Slater determinant.

### 3.6 Localized orbitals

Localized orbitals provide a simple interpretation tool, and are also be used below. For example, for the stationary lowest energy solution, the localized orbitals are just the linear combination of the two lowest energy canonical orbitals with different symmetry,

### 4 Results

### 4.1 Hamiltonian changes slowly with time

Let us first consider systems where the Hamiltonian changes slowly with time. In this case, one can simply use the solutions of the stationary Schrödinger equation, at each moment

Corresponding to this image, it is sufficient to present pictures obtained for different values of the opacity parameter

As expected, our system turns out to be uninteresting. All the methods mentioned above give the same result that can be summarized as "one electron in each of the half-boxes", at all times. Of course, this statement is strict when

### Fig 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, **

The wave function, for

In accordance with it, the perfectly localized orbitals for

ELF takes the maximal value (

Fig. 2f shows the probability of finding one electron between

#### 4.2 Sudden change of the Hamiltonian

#### 4.2.1 Mathematical description

We consider now the opposite extreme, when the modification in time occurs with a jump, from the Hamiltonian with

This expression shows how excited states of the stationary Schrödinger equation for the final Hamiltonian participate to the wave function

#### 4.2.2 Wave function

The evolution of the square of the two-particle wave function with time is presented in Fig. 3. The starting point (

### Fig 3

**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, **

#### 4.2.3 Localized orbitals

Localized orbitals for this process are shown in Fig. 4. The orbitals delocalize into the other half-box (

### Fig 4

**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. **

#### 4.2.4 Density

The change of the density compresses what has been seen above, and some information can be lost, cf. Fig. 5. At ^{10}, Section E2.1.1). At

### Fig 5

**Fig 5
Density, **

#### 4.2.5 "Laplacian" of the density

Instead of the Laplacian of the density we consider again, as suited to the one-dimensional problem, ^{10}, Section 3.2.4). There, although there is just one bond and one would expect a single maximum, the Laplacian of the density shows two maxima.

### Fig 6

**Fig 6
**

#### 4.2.6 Electron localization function

The electron localization function brings in information that is consistent with the information the

### Fig 7

**Fig 7
Electron localization function, **

#### 4.2.7 Maximum probability domain

For interpretation reasons, the maximum probability domains seem to have the simplest structure.Fig. 8 shows the probabilities to find one electron between

### Fig 8

**Fig 8
Probability to find one, and only one electron between **

At

#### 4.2.8 Physical interpretation

Eq.(2) is valid when the change of the Hamiltonian is so fast that the wave function does not have the time to change. After the change, the wave function

where

Fig. 9 shows the squares of the orbitals for times equal to even or for odd numbers of

### Fig 9

**Fig 9
Time-dependent orbitals squared, reduced to the **

The maximum probability domains do not start from an orbital "prejudice", but analyze the total wave function. After the separating wall has vanished, for certain intervals of time, there is a maximum probability domain around the position where the wall has been. Also, by permitting the spatial region to be spatially disconnected, they allow for the description of the quantum phenomenon that a particle can be found in two different disconnected domains.

#### 4.2.9 Comparison to stationary states

The best description of the chemical bond is not necessarily given by a single localized solution even when considering the time-independent case. The simplest example is the H

where ^{13}.)

Another example is given by particles in a ring, or metals, where the localization is not considered to give the best description. Let us assume that for particles in a ring we have found some region, defined by the points

One more example is given by atomic shells. Although, e.g., in an atom like Ne there are four electron pairs, due to spherical symmetry a spatial region defining an electron pair can be oriented in any direction: there are infinitely many equivalent "pair domains". In this case, we consider atomic shells, and only destroying the symmetry fixes the orientation of the pairs.

It is worth to stress that in the time-dependent case discussed in this paper, it is not the symmetry that produces equivalent solutions, but the mixing with excited eigenstates that generates different localization patterns.

Interestingly, Lewis ^{4} had the intuition of the failure of taking his model rigidly. Although desiring to explain polarity, and not the quantum effects discussed here, he writes about "tautomerism, where two or more forms of the molecule pass readily into one another and exist together in a condition of mobile equilibrium".

#### 4.2.10 Period of the cycle

For *via* the transformation

#### 4.2.11 Spatial oscillations

Up to now, an important technical detail was hidden from the discussion, viz., the number of functions

##### 4.3 Explicit solution of the time-dependent Schrödinger equation

##### 4.3.1 Mathematical description

Up to now, we have obtained results in two limiting cases. We would like to know whether the sudden approximation may be relevant. For this, let us consider expand the time-dependent, spatially symmetric wave function as

After substitution of

where the dot above the letter represents the derivative with respect to

Details on solving this equation are given in Appendix D.

In contrast to the treatment before, we cannot start at

##### 4.3.2 Probability evolution

If the change of the opacity parameter

Nevertheless, we discuss how the probabilities evolve with time (see Fig. 10). One of the curves corresponds to the probability of finding one electron in a half-box. The other, to that of finding one electron in the center (equal to that of finding one electron in the disconnected domain that excludes this central region). In Fig. 10, when following the evolution in the central region,

### Fig 10

**Fig 10
Probability to find one, and only one electron in a half-box (between **

Although the probability to find a central MPD is not large at the moment

## 5 Perspectives

The example of two electrons with the same spin shows that when the Hamiltonian changes electron localization may look, for certain time intervals, qualitatively different from what the adiabatic picture presents. In our example, the latter follows that of Lewis, while time dependence brings in quantum delocalization effects. It gives a significant probability of finding an electron in two spatially disconnected regions.

An analogue to the spatially disconnected regions exists for the wave function solving the stationary Schrödinger equation, e.g., when resonant structures are needed to describe the bonding. It can be speculated that phenomena like this play a role, e.g., in charge transfer, in transport properties, also in nano and biological systems.

One should not forget that two particles in a box with an opaque wall do not represent reality, and that no choice of the parameters of the model can compensate for it. However, the simplicity of the model allows us to look at the wave function, and understand better how well, or how badly, the interpretation tools work. Thus, the paper has only two objectives, namely to encourage

● the study of time-dependent processes, as they disclose unexpected situations for chemical bond description, and

● the use of the maximum probability domains that seem well suited for such time-dependent processes.

We finally mention that latter is close to what is already used in time-dependent context, see, e.g. ^{14}, and that limitations of ELF in time-dependent cases has also been noted before ^{5}.

## 6 Dedication

This paper is dedicated to Debashis Mukherjee, who reached his seventies birthday. During the many years of our friendship we spent a long time in discussions on various subjects, including that of the present paper.

### Appendix

### A Solutions of the stationary Schrödinger equation for a particle in a box with an opaque wall

As given in Ref.2 (problems 19, 20), the expressions of the one-particle wave functions, solutions of the stationary Schrödinger equation with potential

and by

where

lying between

Notice that

The eigenvalues are given by

Their dependence

### Fig 11

**Fig 11
Eigenvalues of the stationary Schrödinger equation for a particle in a box with a wall having an opacity increasing with **

### B The expression of the electron localization function for a single Slater determinant

As we are discussing one-dimensional systems, the formula of ELF is slightly different from that generally used. Also, we consider the fully polarized systems, while usually the closed-shell formula is given. We follow the initial choice of the interpretation of ELF ^{11}, viz. related to the curvature of the Fermi hole. It also includes the current contribution ^{15}, as needed when orbitals are complex, as is the case in time-dependent theory ^{12}. For a single Slater determinant, the explicit expression of ELF is

where

while

###
C Connecting the opacity parameter $a$ with time

In order to associate time to the opacity parameter

We can define a constant ^{10}, Section 1.1). For a size of the box given by

We have the freedom to choose

### D Solving the time-dependent Schrödinger equation

In order to solve Eq.(5) a basis has to be chosen. The basis is given by the functions

that correspond to the symmetric eigenfunctions ^{16}.

The expansion in a fixed basis is complicated by the presence of the time-dependent cusp in

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