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 $...
- 10.2k
7
votes
Which expectation values can be determined with KS orbitals?
Since you ask "which" expectation values can be determined this way, instead of "can all" expectation values be determined this way, I'll briefly mention one case where the Kohn-...
- 10.2k
7
votes
Which expectation values can be determined with KS orbitals?
The general answer is: no. Take the total spin, $\hat{\bf S}^2$, for example. Although quantum chemistry programs typically compute the expectation value $\langle \hat{\bf S}^2 \rangle$ using the non-...
- 20.1k
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 ...
- 251
7
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 ...
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 ...
- 15.4k
6
votes
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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 "...
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)...
- 15.4k
6
votes
Constrained optimisation on a hypersphere
I'll propose a slightly different Lagrangian expression:
$$ L = E' + g^T \Delta x + \frac{1}{2} \Delta x^T H \Delta x + \frac{1}{2} \gamma \Delta x^T \Delta x - \frac{1}{2} \lambda (p^T p - k). \tag{1}...
- 3,177
5
votes
Accepted
One-particle coupling coefficient in CI
You made a mistake:
$$(a_{q}^{\dagger} a_{p})^{*} \neq a_{q} a_{p}^{\dagger} $$
Instead it should be:
$$(a_{q}^{\dagger} a_{p})^{*} = a_{p}^{\dagger} a_{q} $$
Hence you can see from the remaining ...
- 520
5
votes
Which expectation values can be determined with KS orbitals?
It cannot hold for all observables. This is a more or less straightforward consequence of the Hilbert space structure of quantum mechanics and DFT plays no role.
Indeed, consider two normalized ...
- 757
5
votes
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How is the equation for the position of a virtual site derived?
The formula (1) in the question:
$$
\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}}...
- 3,177
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 ...
- 3,177
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{\...
4
votes
Accepted
First order variation of the wave function of conduction states
Let's start by unpacking Eq. (1), which we rewrite here:
$$\begin{equation}
(H_{SCF} - \varepsilon_n)\lvert \Delta \psi_n \rangle = -(\Delta V_{SCF} + \Delta \varepsilon_n) \lvert \psi_n \rangle.
\tag{...
- 1,116
4
votes
How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?
As @Jack suggested, try the $\vec{k} \cdot \vec{p}$ method.
The hopping integral $t_{ij}$ is "borrowed" from the Hubbard model. It is the kinetic term in the Hamiltonian that explains ...
- 469
3
votes
Accepted
Adiabatic equation related to the Berry phase for lambda with first order terms
The TD-Schrodinger equation is
$$H(\lambda)|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle$$
So, we just need to plug in \eqref{4} and remove all terms that aren't first-order in $\...
- 15.9k
3
votes
Accepted
Why does the 𝛿 (delta) term exist in the density-density correlation for simple liquids?
While deriving this formula in Ch. 3 of your source, the author references equation 2.5.13,
\begin{equation}
\rho_N^{(2)}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\left\langle\sum_{i=1}^N \sum_{j=1}...
- 950
2
votes
How is the equation for the position of a virtual site derived?
I have posted the same question in Mathematics StackExchange and here is the reply.
Suppose that $d=|\mathbf{d}|$ and $\mathbf{r}_{ij}\ne \mathbf{0}$. Then in the plane of the other three particles ...
2
votes
Cross-post: Matrix elements <n,k|x|n',k'> for Bloch states
If you look at the paper, they cite the work of M. Saitoh in J. Phys. C: Solid State Phys. 5, 914 (1972) for the matrix element.
Saitoh writes in Eqs 6 and 7
$$ X_{j{\bf p},c{\bf k}} = \int {\rm d}^3 ...
- 20.1k
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