Laying the Foundation for Modeling Materials – Part 1: Hamiltonian of the Solid

Part 1 of X. The content of this series of posts is mainly drawn from “Atomic and Electronic Structure of Solids” by Efthimios Kaxiras, in particular Ch.2 on the single-particle approximation (SPA). What follows are my notes, interpretations, and thoughts surrounding the setup of the SPA, following the framework of Kaxiras’ textbook.

Outside of solids that are rather well defined by classical mechanics,* noble solids (e.g. Ne, Ar) and pure ionic compounds (e.g. NaCl), we have an enormous class of materials that are more complex and require a quantum mechanical treatment. In other words, we must consider the impact of the valence electrons on the properties of the solid – how their states influence (and are influenced by) the system. We must do this as the electron nature is no longer insignificant in comparison to the ions in solid systems, and thus treatment as point charges or hard spheres does not capture the solid properties effectively. It incorrectly estimates electron-electron and electron-ion (nuclear) interactions.

By considering the electrons, we are able to procure electronic properties, such as conductivities and electron mobilities, and electronic thermal properties, such as electron-phonon lattice vibrations and scattering. Later on (in future posts) we will discuss these features. At the moment, we must begin by finding the simplest approach to incorporate electronic nature, the quantum phenomena, into our lense for study.

In the single-particle approximation (SPA), we consider the lattice as a set of classical ions and the electrons as individual (single) quantum mechanical particles aimed at reproducing electron behavior. Optical excitations, conductivity, and mechanical properties (e.g. cohesion) are sufficiently described under this approximation.

To try and reach this model, we make an early approximation: ions or nuclei of the lattice in the solid are relatively stationary compared to the electrons, and thus act as a constant external potential on the electrons – this is known as the Born-Oppenheimer approximation (BOA), reasonable to apply due to the immense mass difference between ions and electrons ranging between 3 and 5 orders of magnitude. Conveniently, this allows some simplification in describing solid systems, while we are able to keep the electron-electron interactions which generate an effective potential for approximately capturing many-body features. After constructing our many-body Hamiltonian, however, we will see that key features of this picture are missing in order to capture more complex phenomena such as superconductivity.

Our focus then lies on how the electronic degrees of freedom are affected by the external potential (lattice created) and effective potential (electron/many-body effects). To begin, we have the many-body Schrodinger Equation in matrix form:

which demonstrates that our Hamiltonian (H) operating** on Psi, the state (a.k.a. wavefunction) of our particles based on position of nuclei (R_I) and electrons (r_i), will result in energies (E) for each Psi state. Our Hamiltonian consists of kinetic and potential energy terms – kinetic being a gradient (squared) of the Psi states, potential being the electron-electron and electron-nucleus/lattice interactions relative to distance and charge (and nuclei-nuclei, shown below and to be removed later under our BOA). All terms are summations over all particles, to obtain the total energy of the system being explored.

Kinetic Energy Terms:

[ion/nuclear/lattice kinetic energy] + [electron kinetic energy]

Potential Energy Terms:

[e-e] + [e-n] + [n-n] ; e = electron, n = nuclei

Notes regarding notation: [e-e] is a sum over all “i,j” excluding the “i=j” as it would be the repulsion of an electron in the same position as itself, which is unphysical and would result in a term with division by zero (a discontinuity). Subscripts “I” belong to the nuclei of the lattice, and “i” belong to the electrons (note: mass_electron = mass_e = mass_i, because electron mass is constant and not relative to index “i”).

However, the potential energy terms for solely nuclei and solely electrons double count energies in the summation. For example, with i,j = 1,2,3, the combinations of nuclei/electrons looks like a 3×3 matrix (elements ij, i rows and j columns):

Since all diagonal terms are removed due to the I,i J,j constraint (as it would represent overlapping particles otherwise, and cause a divide by zero), the top and bottom triangular portions are mirror images: 21 = 12, 31 = 13, and 32 = 12, when the particle under the indices is either electron, r, or nuclei, R (when doing the nuclear-electron term, the indices would not be the same). Illustration below:

Red line illustrate the constraint, and the green lines are showing which indices are equal (under either R or r, not when the indices refer to both separately as in the electron-nuclear term)

This situation thus requires a factor of ½ in front of the n-n and e-e potential energy terms (see following full Hamiltonian expressions).

Total Hamiltonian (SPA, no approximations):

Born-Oppenheimer (BOA) removes the kinetic term for nuclei (a.k.a. the lattice) as it is approximately stationary, which results in a constant potential field (i.e. the static lattice) represented by the last term over I,J. As a constant, this term can be ignored for now and added back in later – refer to “Madelung Energy” for determining this value, but for now we hold the (electronic) Hamiltonian as:

Which, by defining that potential interacting with an electron as:

e(I) and e(i) are equivalent, as the electron and proton charge as identical in magnitude – I only express it separately here as a reminder of this fact.

our final many-body electronic Hamiltonian becomes:

because the summation over R_I occurs in the V_ion definition, so the sum in the Hamiltonian shows the summation of r_i over the R_I summation. Formulation is equivalent to showing both indices as V_ion(R_I, r_i), but defining V_ion as we have above is more illustrative of the external (nuclear) potential acting on EACH electron individually, which is what we are truly interested in summing over.

Even with our approximations, we have ignored known characteristics of these particles, in particular the electrons (fermions), which make determination of the wavefunction (Psi) difficult. For instance, electrons of the same spin that exchange position must change the sign of our wavefunction – from general chemistry, this is taught as the Pauli exclusion principle. Additionally, each electron’s state or motion is impacted by all other electrons – this is known as the correlation between the particles. We only capture the stationary potential between electrons, not the impact on that state existing on other electrons. Capturing these two features would begin to make description of properties like superconductivity more feasible (to be discussed later).

Accounting for these properties is done through definition of the wavefunction, under more approximations, to try and capture the proper form of the above many-body wavefunction, and begins with an average approach, otherwise known as a mean-field approximation, starting with an interesting tactic: assuming the many-body wavefunction is a function of non-interacting particles.

To be continued in Part 2: Hartree and Hartree-Fock Approximations


* classical mechanics: newtonian physics, e.g. ball and spring models or hard sphere fluids

** operators: mathematical operations in classical mechanics have quantum equivalents, identical or carrying constants, due to describing particles as waves instead of discrete particles

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