# Questions tagged [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 $K_D[\rho]$...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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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. MĆ¤kinen, Computational Materials Science 2009, 47, 237ā253) and I get stomped by a ...
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### 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: \...
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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 ...
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### 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 ...
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### 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): (H_{SCF}-\epsilon_n)|\Delta \psi_n \rangle = -(\Delta V_{...
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### 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 ...
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