8
votes
Accepted
Ground State energy trick for many-body electronic structure calculations?
There are a few issues with the derivation:
(not necessarily a mistake, but) please note that if you mean the Coulomb Hamiltonian, the RHS of (1) should involve the reciprocals of $|r_i|$ etc., not $...
- 9,076
7
votes
How is the tight binding model derived from the Kohn-Sham DFT energy?
The additional terms come from the Kohn-Sham equations. Everything is detailed quite nicely in W. M. C. Foulkes, R. Haydock, Phys. Rev. B 1989, 39, 12520–12536. Here is a summary of mine:
We start by ...
- 241
7
votes
How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?
How do you get the Hamiltonian formula for transition metal dichalcogenides? How to derivate the expression above (equation (1))?
You can use the $ k \cdot p$ method to derive this effective ...
- 14.9k
6
votes
Accepted
How is the Bloch equation derived?
They start with the time-independent Schrödinger equation:
$$\tag{1}
H|\psi^\alpha\rangle = E^\alpha |\psi^\alpha\rangle.
$$
Then they define $|\psi_0^\alpha\rangle$ to be what they call a "...
- 33.6k
6
votes
Accepted
Analytic solution of Boltzmann equation
Finally, I figure out this question on my own. Put Eq.$(2)$ and $(3)$ into Eq.$(1)$:
\begin{equation}
-e \Re\{\mathcal{E}_a e^{i\omega t}\} (\partial_a f_0+\partial_a f_1)+(\partial_tf_1+\partial_tf_2)...
- 14.9k
6
votes
What is the X in Almlöf and Taylor's "Unified treatment of energy derivatives?"
Question
"Jan Almlöf and Peter R. Taylor (link) provides little further detail, by defining U=exp(X), but not providing details how X can be calculated here."
Answer
$X$ is literally ...
- 33.6k
6
votes
Constrained optimisation on a hypersphere
I'll propose a slightly different Lagrangian expression:
$$ L = E' + g^T \Delta x + \frac{1}{2} \Delta x^T H \Delta x + \frac{1}{2} \gamma \Delta x^T \Delta x - \frac{1}{2} \lambda (p^T p - k). \tag{1}...
- 2,673
5
votes
Accepted
How is the equation for the position of a virtual site derived?
The formula (1) in the question:
$$
\mathbf{r}_s = \underset{\text{term A}}{\mathbf{r}_i}
+ \underset{\text{term B}}{d \cos\theta\, \frac{\mathbf{r}_{ij}}{|\mathbf{r}_{ij}|}}
+ \underset{\text{term C}}...
- 2,673
5
votes
Accepted
How to Derive the Kong-Chakrabarty Mixing Rules
I will focus on deriving equations (3) and (4) from equations (6) and (7). In brief: we use equation (7) to derive $r_j$ as a function of $r_i$, and then differentiate (6) in $r_i$ to derive the ...
- 2,673
4
votes
Accepted
How to formulate the second quantization of Dzyaloshinskii-Moriya interaction?
Let's first break down the symbols we know in that expression:
\begin{align}
\mathbf{\vec{S}} &= \frac{\hbar}{2}\boldsymbol{\vec{\sigma}}\tag{1}\\
&=\frac{\hbar}{2}\left(\sigma_x \boldsymbol{\...
- 33.6k
4
votes
Accepted
First order variation of the wave function of conduction states
Let's start by unpacking Eq. (1), which we rewrite here:
$$\begin{equation}
(H_{SCF} - \varepsilon_n)\lvert \Delta \psi_n \rangle = -(\Delta V_{SCF} + \Delta \varepsilon_n) \lvert \psi_n \rangle.
\tag{...
- 766
4
votes
How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?
As @Jack suggested, try the $\vec{k} \cdot \vec{p}$ method.
The hopping integral $t_{ij}$ is "borrowed" from the Hubbard model. It is the kinetic term in the Hamiltonian that explains ...
- 459
3
votes
Accepted
Adiabatic equation related to the Berry phase for lambda with first order terms
The TD-Schrodinger equation is
$$H(\lambda)|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle$$
So, we just need to plug in \eqref{4} and remove all terms that aren't first-order in $\...
- 15.2k
3
votes
Accepted
Why does the 𝛿 (delta) term exist in the density-density correlation for simple liquids?
While deriving this formula in Ch. 3 of your source, the author references equation 2.5.13,
\begin{equation}
\rho_N^{(2)}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\left\langle\sum_{i=1}^N \sum_{j=1}...
- 890
2
votes
How is the equation for the position of a virtual site derived?
I have posted the same question in Mathematics StackExchange and here is the reply.
Suppose that $d=|\mathbf{d}|$ and $\mathbf{r}_{ij}\ne \mathbf{0}$. Then in the plane of the other three particles ...
2
votes
Cross-post: Matrix elements <n,k|x|n',k'> for Bloch states
If you look at the paper, they cite the work of M. Saitoh in J. Phys. C: Solid State Phys. 5, 914 (1972) for the matrix element.
Saitoh writes in Eqs 6 and 7
$$ X_{j{\bf p},c{\bf k}} = \int {\rm d}^3 ...
- 18k
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