# Questions tagged [derivations]

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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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### 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, ...
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### 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 ...
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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 ...
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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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### 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 ...
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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}...
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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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### 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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### 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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### 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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### 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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### 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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