17

Both Hartree and Hartree-Fock have a self-consistent field method, as does density functional theory, too; the difference is just that Hartree theory uses a bosonic wave function (symmetric with interchange of particles), while Hartree-Fock uses a fermionic wave function i.e. a Slater determinant that is the correct one for electrons. In both cases, the ...


16

Following your arguments, we would see also a 'violation' of the Heisenberg uncertainty principle (HUP) in single-particle quantum mechanics, e.g. in a Hamiltonian of a particle in an external potential: $$ H = \hat{p}^2/2m + V(\hat x) \quad .\tag{1}$$ But writing down such a Hamiltonian does not mean that we know the position or momentum (or both) of this ...


14

Kieron Burke and co-workers have shown that in many cases one can get better results by using the HF density as input to a DFT-XC evaluation of the energy, as opposed to using the DFT-XC to generate a self-consistent density. This has been termed density-driven error, as opposed to functional-error. SE is especially problematic in cases where an electron is ...


13

It comes down to the fact that HF and KS both are variational methods. This short article by Julien Toulouse gives a great description of ways to compute static/dynamic response properties. I'll just summarize the relevant portion. We can compute derivatives of the energy with respect to any variable $x$ as: $$\frac{dE}{dx}=\frac{\partial E}{\partial x}+\...


12

This question is a bit ill-defined: what do you mean by "the self-consistent field procedure"? If you mean the original Roothaan procedure, then the question makes sense, but it is uninteresting: nobody uses the Roothaan procedure, since it usually doesn't converge, and you need to do something smarter like use damping or other convergence acceleration ...


12

The procedure to find the Hartree equations and the Hartree-Fock equations is very similar, we have to minimize the expectation value of the Hamiltonian under the orthonormalization constraint. However, both methods differ in the form of the wavefunction. Hartree Method In the Hartree method, the total wavefunction is a Hartree product: $$ \Phi(\mathbf{r}...


12

You're right about the ability to change the initial guess repeatedly until you get the lowest energy, and this is how it's done in software like MOLPRO which don't offer "stability analysis". However in software like GAUSSIAN and CFOUR, you can do something called stability analysis, which is described for example in the GAUSSIAN documentation ...


12

Indeed, due to the uncertainty relation, the electrons in a molecule do not have a definite potential energy, nor do they have a definite kinetic energy (yet they have well-defined expectation values of potential energy and kinetic energy). But, assuming that the molecule is in an energy eigenstate, the two uncertainties exactly cancel out, and gives an ...


11

There are a lot of different methods, but a good starting point to understand how this might be done is the method of steepest descent. First we take the generalised eigenvalue expression you cite and rearrange to express the energy expectation value as $$E=\frac{\phi^\dagger \mathrm{H}\phi}{\phi^\dagger \mathrm{S}\phi},$$ where I've generalised to allow ...


11

Your Octave code is trying to do the integral by quadrature, which makes very little sense since it will have a huge problems with the cusp. Since this is a one-center problem, the best approach is to use the Legendre expansion for $|r_1-r_2|^{-1}$, which decomposes the interaction into a radial part and an angular part: $r_{12}^{-1} = \frac {4\pi} {r_>} ...


11

GHF: Generalized Hartree Fock In Restricted Hartree-Fock (RHF), the molecular orbitals are constructed as pairs, with a single spacial function being used to describe both an $\alpha$ and $\beta$ spin electron. Unrestricted Hartree-Fock (UHF) lifts this requirement, forming a unique set of MOs for $\alpha$ and $\beta$ spin. Generalized Hartree-Fock takes ...


10

DHF: Dirac-Hartree-Fock (or "Dirack-Fock") the DHF (Dirac-Hartree-Fock) or Dirac-Fock is the SCF method based upon four-component spinors (simply four-spinors), because of the four-component Dirac-Coulomb(-Breit/Gaunt) Hamiltonian. The 4-spinors decribe both positive - electronic - solutions as well as negative, or "positronic" solutions....


10

Just like geometry optimization, there is no practical way to be 100% sure that you have the global minimum of SCF solutions. But there are checks you can do to make sure that the SCF solution you got is a reasonable minimum. One of them is checking the electronic hessian at the SCF solution and determining the lowest eigenvalues. If one or more negative ...


9

Only the first line is correct, as long as the AO basis is not orthogonal. The point is: the AO $\leftrightarrow$ MO transformations of density matrices and Fock matrices require different formulas. For the Fock matrix the formulas are \begin{align} F_{MO} &= C^T F_{AO} C \tag{1}\\ F_{AO} &= SC F_{MO} C^TS \tag{2} \end{align} While for the density ...


9

RHF: Restricted Hartree-Fock / RKS: Restricted Kohn-Sham Restricted Hartree-Fock (RHF) is a self-consistent field approach: a mean-field approximation to the electronic, non-relativistic Schrödinger equation. The electron-electon interaction is modeled as a field influencing all electrons in the system, which are otherwise independent particles. Their "...


9

The short answer is: it is the matrix representation of the Fock operator in the given basis set, in this case, the atomic orbital (AO) basis. The Fock operator itself is a mean-field, independent particle approximation to the electronic Hamilton operator of the system (with other approximations beyond the scope of this Q&A). The rows/columns (the matrix ...


9

A common textbook statement of the uncertainty principle is: $$\tag{1} \Delta x \Delta p \ge \frac{\hbar}{2}, $$ where $\Delta x$ and $\Delta p$ are the uncertainties in measurements of the position $x$ and momentum $p$ respectively. What mathematically is an "uncertainty"? In this case it's the standard deviation: \begin{align} \tag{2} \Delta x &...


8

I guess this will go in the answer slot, it is a bit long for a comment. DFT typically has quite a bit less spin contamination than HF (attributed to the inclusion of correlation). One issue however is that DFT tends to favor electron delocalization, and conveniently HF tends to the opposite. This is part of why hybrid DFT methods get improvements, the ...


8

This can be solved analytically, a complete solution can be found here To refrain from rewriting the entire derivation I will only say that you need to integrate over all 3 dimensional degrees of freedom for both electrons, so TAR86 is correct. In the derivation at the link, the distance between the electrons ($\mid r_1 - r_2 \mid \equiv r_{12}$) is better ...


7

A couple of comments perhaps to clarify some points. The first Hohenberg-Kohn theorem states that the one-electron density determines uniquely the energy, for a non-degenerate ground state. The second H-K theorem states that this density can be obtained by employing the variational principle, i.e. the density is obtained by minimizing the energy as a ...


7

As Nike has already mentioned, Slater determinants occur in KS-DFT in much the same way that they do in Hartree-Fock as a way to combine a set of one-electron orbitals into an antisymmetric many-electron wavefunction. However in DFT, as the name suggests, we are typically more interested in the density. The Hohenberg-Kohn theorems show that DFT can, in ...


7

From the Wikipedia article on Kohn-Sham equations: "As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals." So the answer to your question: "Are slater determinants found in practice in Kohn-Sham Density Functional theory?" is ...


7

Short answer: don't do it. While optimizing the exponents in a basis set for your system is possible in principle, the optimization is very expensive. You may also end up ruining the accuracy of your properties, since the basis set may become biased. Basis sets of various quality are routinely available in the literature. Instead of reoptimizing the ...


7

The normalization constant is built into the Roothaan-Hall equation. Namely, when you derive the Hartree-Fock energy expression from the wave function, you integrate out all the orbitals that don't "touch" the Hamiltonian, and the permutations of the orbitals "kill off" the normalization constant. Now, when you have the energy expression, ...


7

This is just the Gaussian cube format. It's essentially a voxel dump of the wave function evaluated on a grid. You can find some documentation at http://paulbourke.net/dataformats/cube/ and https://h5cube-spec.readthedocs.io/en/latest/cubeformat.html . Most electronic structure programs are able to generate Gaussian cube files. Several molecular viewers can ...


7

If by "the SCF method" you mean the simple SCF, the answer is: no, usually it does not converge (unless the problem is very simple; basically the gap is huge, so that the system does not respond very much to potentials). The damped SCF problem on the other hand does converge, for damping parameters small enough (unless you run into fractional occupations). ...


7

I don't think analog computers are quite at the level they need to be yet, but there are people working to make them applicable to matter modeling applications. Dr. Rahul Sarpeshkar at MIT developed simple circuits to model a particular pair of cellular protein production processes and mimic the differing pathways to the same result. They cite the continuous ...


7

For canonical MOs, it is expected that the off-diagonal elements of the described product with the Fock matrix are zero (within reasonable accuracy) because they are the result of the diagonalization of the Fock matrix. However, canonical MOs are not the only possible choice. One can "rotate" the MOs to minimize certain metrics in order to obtain, ...


6

Formally, the Fock matrix is the density matrix derivative of the Hartree-Fock or Kohn-Sham energy functional. The Fock matrix returned by PySCF is in the atomic-orbital basis, which is actually the same as the molecular basis. If the orbitals satisfy the self-consistent field equations, then in the molecular orbital basis the Fock matrix is diagonal. See ...


6

For simplicity, I will stick to the restricted Hartree-Fock level of theory since the question of canonical and semi-canonical orbitals already exists there. Let's remember the SCF equations: ${\bf F C} = {\bf SCE}$, where ${\bf F}$ and ${\bf S}$ are the Fock and overlap matrices, with ${\bf C}$ the orbital coefficients and ${\bf E}$ the corresponding ...


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