Many methods exist to compute the large scale dynamics of systems in their electronic ground state, such as conventional [1] and reactive force fields [2], and *ab initio* molecular dynamics [3]. However, we often wish to compute the large scale excited electron dynamics of systems with energies hundreds of electron volts above the ground state, where a multitude of adiabatic states exist, and where condensed materials can coexist with plasmas or highly excited electrons. Few existing methods are fast, accurate, and general enough to satisfy this need.

We introduce an electron force field (eFF) that with only three universal parameters can compute the excited electron dynamics of systems containing hydrogen, helium, lithium, beryllium, boron, and carbon. In our model, nuclei are represented by point charges and electrons by spherical Gaussian wave packets with variable position and extent. Geometries are reproduced well, and energies are calculated with sufficient accuracy so that we can simulate the excited electron dynamics of matter at extreme conditions. We use as examples the temperature dissociation and ionization of high-pressure deuterium, and the Auger fragmentation of hydrocarbons induced by removal of core electrons.

The use of sums of Gaussians to approximate wavefunctions needs no introduction, as Gaussians are prominent in practically every modern *ab initio* method today [4], due to the simplicity of computing integrals of said functions. The use of *single* Gaussian wave packets to represent quantum particles may be less familiar to the reader, and the promise that such a drastic approximation might yield quantitatively accurate quantum dynamics of nuclei, nucleons, or even electrons has motivated research on this topic for the last several decades.

In 1975, Heller [5] demonstrated that the equations of motion for ``thawed'' Gaussian wave packets in locally harmonic potentials had a particularly simple form, and used them to compute the quantum dynamics of colinear . He later pioneered use of time-dependent methods to obtain spectroscopic data, for example computing the photodissociation cross-section of methyl iodide [6], and the three-dimensional photodissociation dynamics of ICN [7]. In these cases, the quantum particles were nuclei moving in a parameterized potential where electrons were considered only implicitly.

Computing explicit interactions of indistinguishable fermions such as nucleons or electrons is more difficult than computing interactions between nuclei, because the overall electronic wavefunction must satisfy an antisymmetry principle, which specifies that interchanging any two fermions causes the sign of the wavefunction to change. The simplest function that satisfies this requirement is the antisymmetric sum of N! product wavefunctions; if we assume pairwise electrostatic interactions, evaluating the energy of such a wavefunction requires operations. In contrast, computing the energy of a Hartree product wavefunction, which does not satisfy the Pauli principle, requires at most operations ( for electrostatics, and N for kinetic energy).

For practical molecular dynamics, we would like energy evaluation to have better scaling than , which leads to two questions:

- Given N electrons represented by single Gaussian functions, if we take the wavefunction to be the fully antisymmetrized combination of these functions, with the known cost for energy evaluation, do we get a reasonably correct description of molecules?
- We define the antisymmetrization energy or Pauli energy as the difference in energies of an antisymmetrized wavefunction (which satisfies the Pauli principle) and a product wavefunction (which does not). Can we approximate this energy with an expression that is faster to evaluate than ?

The first question was answered by Frost [8] in 1964, with his development of the floating spherical gaussian orbital method (FSGO). In FSGO, the wavefunction is an antisymmetized set of floating Gaussian orbitals
. The energy is simply the combined kinetic and electrostatic energy of this wavefunction; since no adjustable parameters are used, the method is considered fully *ab initio*. Despite the simplicity of the basis functions, Frost found good geometries for molecules like lithium hydride, beryllium dihydride, first and second row hydrides, and hydrocarbons. He concluded that single floating Gaussians have a variational flexibility comparable to larger sums of nuclear-centered fixed-size Gaussians of the sort used in traditional *ab initio* calculations.

The second question was answered in the late 70s and over the next two decades with the development of Pauli potentials by Wilets et al. [9], Kirschbaum and Wilets [10], Hansen and McDonald [11], Dorso et al. [12,13], Boal and Glosli [14,15], and Klakow et al. [16], which approximated the antisymmetrization energy with an pairwise sum between electrons. Typically these potentials exclude some region in position-momentum phase space, so that electrons are well-separated in position and momentum over a wide range of conditions [17]. These potentials have been used to study nuclear collisions and reactions [18], proton stopping by molecular targets [19], as well as hydrogen plasma dissociation and ionization [16,20].

However, it has been difficult to find a Pauli potential that is accurate enough to keep molecules with larger Z atoms stable, let alone have correct energies and geometries. The most accurate Pauli potential to date, used by Klakow [16] to describe hydrogen plasma, can compute the interaction between electrons of different sizes -- essential for capturing changes in bonding during chemical reactions -- yet it causes lithium hydride to be unbound and the valence electrons of alkanes to collapse onto their cores. Kirschbaum's potential [10] has been applied to create stable atoms with Z up to 94 [21] which have a shell structure, but the potential does not describe the structure of valence shells with enough accuracy to form reasonable bonds between atoms.

There is a need for a Pauli potential with improved accuracy for molecular systems. In the current work, we have developed a Pauli potential that scales as and is applicable to a large range of molecules, including hydrocarbons, which makes it possible to study the excited dynamics interactions of many kinds of bonds -- covalent, ionic, multicenter -- in many phases of matter -- solid, liquid, gas, plasma. In addition to computing the excited state dynamics of high-energy systems, we have validated eFF against a range of simple ground state molecules, with an aim towards highlighting its strengths and particularly its weaknesses, so that it may be improved further in the future.

Although the current eFF contains only one parameterized term -- the Pauli potential -- we call it a force field because we expect that future improvements will hinge on adding physically motivated terms describing more subtle interactions between electrons and nuclei. Such a force field may open the door to truly practical quantum dynamics on large scale atomic and molecular systems.

The chapter is organized as follows: first we discuss the energy expressions of eFF, show how both hydrogen atom and hydrogen molecule are stable, and give a motivation for our form of the Pauli potential. Then we test how well eFF describes ground state systems, with particular attention to the conformers of hydrocarbons, and the effects of breaking hydrocarbon bonds. We also test systems that include lithium, beryllium, and boron; these contain ionic and/or electron-deficient multicenter bonds, and eFF describes them reasonably well. Having validated eFF against ground state systems, we discuss its application to matter at extreme conditions, using as examples the dissociation and ionization of warm dense hydrogen, and the dynamics of the Auger process in hydrocarbons.