Questions tagged [derivations]

Questions about mathematical derivations.

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Cutoff length for vdW correction in DFT calculation

In classical MD simulations of lattice structures, whenever we are trying to incorporate vdW corrections to the atomic force calculation, we need to set up cutoff length beyond which the vdW ...
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5 votes
1 answer

Ground State energy trick for many-body electronic structure calculations?

I am an outsider to this field, so I am not sure about the validity of my work below. Let us define the following Hamiltonian from DFT: $$ \tag{1}H_{ij} \psi_{ij} \equiv (-\frac{\hbar^2 \nabla_i^2}{2m}...
8 votes
1 answer

Analytic solution of Boltzmann equation

This question is related to nonlinear Hall effect proposed in this paper. The Boltzmann equation in the electric field under relaxation time approximation is: $$-e E_a \partial_a f+\partial_t f=\dfrac{...
  • 14k
6 votes
1 answer

How to formulate the second quantization of Dzyaloshinskii-Moriya interaction?

The Dzyaloshinskii-Moriya interaction (DMI) existing in the interface of the ferromagnetic insulator and the metal with strong spin-orbit coupling (SOC) is shown below. Mathematically, it can be ...
  • 14k
15 votes
1 answer

First order variation of the wave function of conduction states

The first order variation of the wave function $\Delta \psi_n$ is obtained by standard perturbation theory (Eq. 25 of ref 1): \begin{equation} (H_{SCF}-\epsilon_n)|\Delta \psi_n \rangle = -(\Delta V_{...
12 votes
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Orbital magnetization in periodic solids

The orbital magnetization in periodic solids has been nicely described by the so-called modern theory of magnetization [1,2,3,4]. $$\tag{1} \mathcal{M}_{orb} = -\frac{1}{2} \Im \sum_{n,\mathbf{k}} f_{...
8 votes
1 answer

How is the Bloch equation derived?

I am reading a textbook on many-electron theory, which gives a Bloch equation in its generalized form: $$\tag{1} [\Omega,H_0]P=Q(V\Omega-\Omega V_{eff})P, $$ where $P$ denotes the projection from ...
  • 3,609
7 votes
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Deriving relations for a hard sphere phase diagram [closed]

In Torquato's book "Random Heterogeneous Materials", he has written: $$\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+})\tag{1}$$ where $g_2(D^+)$ is the contact value from the right-side of the ...
  • 3,947
11 votes
2 answers

How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?

TMDs are transition metal dichalcogenides and have the chemical formula MX$_2$ where M is the transition metal and X is the chalcogen. An example of a TMD is MoSe$_2$. I would like to demonstrate that ...
12 votes
1 answer

Adiabatic equation related to the Berry phase for lambda with first order terms

Consider the following derivation in David Vanderbilt's book "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" (2018, ...