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Electron force field makes large scale simulations possible.

To understand how electron force fields can help us compute approximate excited state dynamics quickly, we make an analogy to traditional force field methods for calculating ground state dynamics (Figure 3.2).

Figure 3.2: Electron force field makes feasible simulations of large scale excited electron dynamics.

In the ground state, we assume that nuclei are well-localized, so that they can be represented classically; and that since electrons are much lighter than nuclei, the electron wavefunction and energy is a parameteric function of the nuclear coordinates (Born-Oppenheimer approximation [4]):

    $\displaystyle \mathcal{H}_{elec}(\mathbf{R}) \psi(\mathbf{r}) = E_{elec}(\mathbf{R}) \psi(\mathbf{r})$ (3.1)
    $\displaystyle E_{total}(\mathbf{R}) = \sum_{i<j} \frac{Z_{i} Z_{j}}{R_{ij}} + E_{elec}(\mathbf{R})$ (3.2)

where $ \mathbf{R}$ are the nuclear coordinates and $ \mathbf{r}$ are the electron coordinates.

To compute how the nuclei move over time, we can solve the time-independent electronic Schrodinger equation at each time step, compute forces on the nuclei, get new positions and velocities by integration, and repeat. The forces are calculated using the Hellman-Feynman theorem [5], which states that the force on each nucleus in the presence of a normalized wavefunction $ \psi$ is the sum of the electric field from the other electrons and nuclei, as well as a Pulay force [6,7] that goes to zero if $ \psi$ is an eigenfunction of $ \mathbf{H}$:

$\displaystyle \mathbf{F}$ $\displaystyle =$ $\displaystyle -\mathbf{\nabla} E = -\nabla \braket{\psi\vert\mathcal{H}\vert\psi}$  
  $\displaystyle =$ $\displaystyle \underbrace{-\braket{\psi\vert\mathbf{\nabla \mathcal{H}}\vert\ps...
... - \braket{\psi\vert\mathcal{H}\vert\mathbf{\nabla}\psi}}_{\mathbf{F_{Pulay}}}.$ (3.3)

However, this method, called Born-Oppenheimer molecular dynamics [8], is slow, because it requires a quantum mechanics calculation at each step. We may speed this up by only partially optimizing the wavefunction at each step, as in the Car-Parinello approach [9], or by using particularly efficient density-functional theory methods [10], but in practice ab initio based MD simulations remain limited to hundreds of atoms over picoseconds [3].

Is there any way to get around solving Schrodinger's equation at each step? After all, the Born-Oppenheimer approximation states that the total energy should be a function of nuclear coordinates alone. In fact, the most common way to perform dynamics on large systems over long times is to use a force field, an approximate energy expression that is a function of nuclear coordinates. A force field typically combines covalent terms between and about bonds with pairwise noncovalent terms between atoms [12,1,2]

$\displaystyle E_{total}(\mathbf{R}_{nuc})$ $\displaystyle =$ $\displaystyle \sum_{i<j} E_{bond}(R_{ij}) + \sum_{ijk} E_{angle}(\theta_{ijk}) + \sum_{ijkl} E_{dihedral}(\phi_{ijkl})$  
    $\displaystyle + \sum_{i<j} \frac{q_{i} q_{j}}{R_{ij}} + \sum_{i<j} E_{vdW}(R_{ij})$ (3.4)

The ability to parameterize high-accuracy quantum and experimental data into a lower-accuracy nuclear potential is fundamental to being able to simulate systems ranging from homogeneous [15] to hetergeneous [16] catalysts, from amino acids to proteins with solvent effects included [17]. In most force fields, the partitioning of energy terms is physically motivated, and it requires some artistry to determine the functional forms needed to reproduce a broad range of chemical phenomena. The reward for undertaking the laborious procedure of force field development is a function that can be orders of magnitude faster to evaluate than quantum mechanics.

In an electron force field, we consider the situation where electrons are not in the lowest energy state: the bandgap may be small, the temperature may be high, a current may be flowing, light may have excited electrons, or free electrons may be present. In many of these cases, the Born-Oppenheimer approximation no longer holds, and we write the force field energy as a function of both nuclear positions and a reduced set of electron parameters. Which electron parameters to include is a balancing act -- too few, and our description will be inadequate to explain chemistry; too many, and the resulting function will be as expensive to evaluate as quantum mechanics.

Following a systematic investigation into the terms needed to capture a broad range of chemical phenomena, we have developed a force-field expression that is a function of nuclear positions $ \mathbf{R_{i}}$, average electron positions $ \mathbf{x_{i}}$, and average electron sizes $ s_{i}$:

    $\displaystyle E_{total}(\mathbf{R}, \mathbf{x}, \mathbf{s}) = \sum_{i\in\mathit{elec}} E_{kinetic}({s}_{i})$  
  $\displaystyle +$ $\displaystyle \sum_{i,j\in \mathit{nuc}} \frac{Z_{i} Z_{j}}{R_{ij}}
+ \sum_{\su...
..._{i}, Z_{j})
+ \sum_{i,j\in \mathit{elec}} V_{elec-elec} (x_{ij}, s_{i}, s_{j})$  
  $\displaystyle +$ $\displaystyle \sum_{i,j \in \uparrow\uparrow \mathit{elec}} E_{\mathrm{exch}}(x...
...}^{\mathrm{p-like}}(x_{ik}, x_{jk}, \theta_{ikj}, s_{i}, s_{j}, Z_{k})\nonumber$  
  $\displaystyle +$ $\displaystyle \sum_{i,j \in \uparrow\downarrow \mathit{elec}} E_{\mathrm{corr}}(x_{ij}, s_{i}, s_{j})$ (3.5)

where all pairs are counted once, and the nuclei have charge Z. The energy is a sum of electronic kinetic energies (larger for small electrons); screened electrostatic interactions between all pairs of nuclei, electrons, and electrons and nuclei; repulsive Pauli exchanges between same spin electrons; attractive Pauli exchanges between same spin electrons near the same nucleus; and attractive correlations between opposite spin electrons.

We emphasize similarities and differences between eFF and traditional force fields. Our electron force field is like traditional force fields in that it is fast, and all the methods used to speed up evaluation of traditional force fields, such as neighbor lists [18], multigrid Poisson solvers [19], particle mesh Ewald [20], and so on, can be used to make the electron force field faster as well. In that respect it advances our goal of making large-scale excited electron dynamics simulation practical.

However, the electron force field is different from traditional force fields in that properties such as bonding, hybridization, lone pairs, bond geometry preferences, steric effects, transition state energies, charge distributions, number of electrons in valence shells, spin multiplicity effects, and ionization potentials all appear as emergent properties of the interactions between nuclei and electrons. It is our hope that features of our force field may be incorporated into traditional force fields, so that they can be made more general while requiring fewer parameters.

We note finally that force field development serves a pedagogical as well as a practical purpose. Force fields break interaction energies into components that are both convenient to calculate and easy to understand in an physical way: terms like bond stretching, van der Waals interactions, electrostatics, and so on. The price paid for the accuracy of high-level theory is often a loss in our ability to analyze results; force fields are a way to recapture this understanding [21].

next up previous contents
Next: Wave packet molecular dynamics Up: The electron force field, Previous: Simulating excited electron dynamics   Contents
Julius 2008-04-29