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Questions tagged [derivations]

Questions about mathematical derivations.

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Derivation of the Dirac exchange-energy formula

So I have been reading the textbook "Density Functional Theory of Atoms and Molecules" by Parr and Yang, and in chapter 6, their derivation of the Dirac exchange-energy functional $K_D[\rho]$...
user392401's user avatar
9 votes
3 answers
115 views

Which expectation values can be determined with KS orbitals?

Suppose $A$ is some hermitian operator and $\Psi$ is a many body state function of a many-body hamiltonian $H = T + U + V$, where $U$ is electron-electron interaction and V is electron-nuclear ...
Mikke Mus's user avatar
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5 votes
2 answers
156 views

How is the equation for the position of a virtual site derived?

Cross posted on Math SE I am trying to understand virtual sites in MD simulations, and I came across this configuration: Here, coordinate $\mathbf{s}$ represents the virtual site, which is formed by ...
Vasista's user avatar
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6 votes
1 answer
81 views

What is the X in Almlöf and Taylor's "Unified treatment of energy derivatives?"

I have been studying the possible methods for basis set optimizations. One notable paper is "Energy-optimized GTO basis sets for LCAO Calculations. A Gradient Approach" by Knut Faegri Jr. ...
John's user avatar
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5 votes
1 answer
215 views

Constrained optimisation on a hypersphere

I am currently trying to implement a GS2 (Gonzalez-Schlegel second order) IRC algorithm in a python code. I am following the original paper ref(1). The main problem is in the constrained optimisation ...
S R Maiti's user avatar
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4 votes
1 answer
84 views

Why does the 𝛿 (delta) term exist in the density-density correlation for simple liquids?

Cross-posted from the PhysicsSE. I am reading Theory of Simple Liquids by Hansen and McDonald, and they in chapter 3, they describe the density-density correlation for a simple liquid in the grand ...
megamence's user avatar
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5 votes
1 answer
109 views

How to Derive the Kong-Chakrabarty Mixing Rules

Background In the world of atomistic modeling with classical force fields, one is often given a force field defined by like interactions (e.g. argon-argon interactions). If one is working with a ...
Hayden S's user avatar
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1 answer
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How is the tight binding model derived from the Kohn-Sham DFT energy?

I am following along a publication titled "Density-functional tight-binding for beginners" (P. Koskinen, V. Mäkinen, Computational Materials Science 2009, 47, 237–253) and I get stomped by a ...
Roberto's user avatar
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5 votes
1 answer
103 views

Cross-post: Matrix elements <n,k|x|n',k'> for Bloch states

Cross posted at Physics.SE I believe this is just elementary QM, but I'm getting awfully confused. The question is drawn from this paper on Wannier-Stark localization (but is self-contained). Let: \...
dsfkgjn's user avatar
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Cutoff length for vdW correction in DFT calculation [closed]

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 ...
Lonitch's user avatar
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6 votes
1 answer
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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}...
More Anonymous's user avatar
8 votes
1 answer
205 views

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{...
Jack's user avatar
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6 votes
1 answer
415 views

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 ...
Jack's user avatar
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15 votes
1 answer
271 views

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_{...
Xiaoming Wang's user avatar
14 votes
0 answers
252 views

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_{...
Xiaoming Wang's user avatar
8 votes
1 answer
195 views

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 ...
Paulie Bao's user avatar
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7 votes
0 answers
120 views

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 ...
megamence's user avatar
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11 votes
2 answers
632 views

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 ...
Carmen González's user avatar
12 votes
1 answer
173 views

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, ...
Carmen González's user avatar