24

First note that the acronym DFA I used in my comment originates from Axel D. Becke paper on 50 year anniversary of DFT in chemistry: Let us introduce the acronym DFA at this point for “density-functional approximation.” If you attend DFT meetings, you will know that Mel Levy often needs to remind us that DFT is exact. The failures we report at ...


24

There is nothing trivial about MCSCF calculations because it is hard to predict a priori how long a calculation will take. There are well-defined equations for calculating how many determinants $$ D(n,N,S) = \binom{n}{N/2+S} \binom{n}{N/2-S} $$ or configuration state functions (CSFs) $$ D(n,N,S) = \frac{2S+1}{n+1} \binom{n+1}{N/2-S} \binom{n+1}{N/2+S+1} $$...


13

It is generally recommended not to use a cc basis set with a DFT method (and I guess conversely, a basis set aimed at DFT should not be used with a coupled cluster method). This statement glosses over some specifics that might be important. There is nothing technically wrong with using correlation-consistent or ANO basis sets with DFT, unless the basis is ...


13

No. The reason for this is not to be found in the excitations, but in the evaluation of the method, i.e. the working equations. $$%Introducing some shortcuts \require{cancel} \newcommand{\op}[1]{\mathrm{#1}} %\op{H} \newcommand{\bracket}[2]{\left\langle#1\middle|#2\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\ket}[1]{\left|#1\right\...


10

I think you are maybe confusing how dynamical and static correlations are treated with different method. Also CASSCF by itself is not a multi-reference method. CI in general is able to describe both dynamical and static correlation (FCI does at least). What is treating dynamical correlations (but not static) is the truncation scheme using different degrees ...


10

When $N_{\alpha} = N_{\beta}$, a restricted solution of Roothaan equations is a solution to the unrestricted Pople-Nesbet equations. This restricted solution always exists and necessarily results if an initial guess $P_{\alpha} = P_{\beta}$ is used. [...] In seeking this second solution [the unrestricted one], it is imperative that an initial guess $P_{\...


10

Diagonalization of the core Hamiltonian provides usually not the best guess for the SCF procedure to say the least, and thus, by default Gaussian uses a more sophisticated guess obtained by diagonalizing the Harris functional (Guess=Harris). With this default guess one get the same energy as OP reported: SCF Done: E(RHF) = -107.495842181 A.U. after ...


9

Your question actually contains two linked inquiries. The proof itself depends on the nature of the Fock-operator to which I'll have a few words afterwards. For the proof itself it is only important, that the Fock matrix is diagonal. Let's restate Brillouin's theorem: In case of the Hartree-Fock (HF) reference determinant $\Phi_0$, the Hamiltonian matrix ...


9

First off, I would like to note that you wrongly interpreted the very first equation. Below I quote the relevant part of the paper: In the Hartree-Fock (HF) approximation for, say, the $\ce{Na}$ atom, the ground-state wave function has the form $$ \Psi = \mathcal{A} (\Phi_{\mathrm{core}} \phi_\nu X) \, , \tag{1} $$ where $\Phi_{\mathrm{core}}$ ...


8

Because in fact it is appropriate. In most cases there is not a huge difference (quality/efficiency) among basis set families. For example Dunning (cc) basis sets work reasonable well for DFT, and Alrichs's (def2) are ok for basis set extrapolations. There could be many reasons for the choice: Diffuse augmentation functions were designed together with ...


8

The experimental equivalent of your question would be: what kind and level of impurity am I going to accept in my experiment? In terms of basis sets: in principle you keep increasing the complexity of your basis set until you reach convergence in terms of electronic energy. What you consider to be converged depends of course on your preferences. With ...


8

The convention used by many is that ab initio refers solely to wave-function based methods of various sorts and that first principles refers to either wave-function or DFT methods with little approximation. I can't find a citation at the moment, but I know this convention is fairly widely used in, e.g., J. Phys. Chem. journals. The IUPAC gold book doesn't ...


8

Actually there is a mistake in the analtical expression in Cook's Book. On his web page he has a pdf with the corrected verison http://spider.shef.ac.uk/ Maybe this solves your problem, but I would also recommend to implement the Obara-Saika Scheme or rys-Quadrature since they are really much more efficient. If your are programming in Python, you might ...


7

The distinction between static and dynamic correlation is not well-defined.1 The distinction is only sensible with respect to a single-particle picture, i.e. viewing the many-electron wavefunctions as built up from single electrons. Let's start with some notes about configuration interaction (CI). The idea of configuration interaction is to express the ...


7

I am not aware of any existing Fortran code for direct numerical quadrature of this problem, but it is worth pointing out that Mathematica can perform this integral symbolically: Integrate[t^(2 n) Exp[-x t^2], {t, 0, 1}] (* 1/2 x^(-(1/2) - n) (Gamma[1/2 + n] - Gamma[1/2 + n, x]) *) where the $\Gamma$ function can be computed by exponentiating easy-to-find ...


6

The Hartree-Fock method minimizes the energy by diagonalizing the Fock matrix, therefore by definition we have \begin{equation} \langle \chi_i|f|\chi_j\rangle = 0, i \neq j \end{equation} where $\chi_i$ are called canonical orbitals (whether you use spin or spatial orbitals is not important here). In other words the "nice enough" orbitals are ...


6

My advice is to implement the Obara-Saika recurrence formulae that are outlined in "Molecular Electronic-Structure Theory" by Helgakar, et al. I would stick with Cartesian functions, since a) they are easier and b) spherical harmonics don't matter for molecules anyway. I did this years ago (in Mathematica) when I was in a similar place -- having completed ...


6

My previous answer was downvoted (-3!), but I maintain that in the particular case where there are only two electrons, CCSD is equivalent to CISD and Full-CI. If the discussion was about asking if for a one-electron system Coupled Cluster gives an upper bound to the energy, everyody would agree that it is true. I agree with all that was said in the top ...


5

In the most general case, CC can be understood simply as a prescription for a trial wave function (ansatz) that uses the exponential excitation operators. This ansatz can be then optimized variationally, and this is variational CC.[1] It is much more expensive than the common CC that uses another approximation (besides the ansatz), which linearizes the CC ...


5

My suggestion would be to use another existing code and run the calculation. For example, if I do an HF/STO-3G calculation on $\ce{H2O}$, I get: $$E_\mathrm{H_2O}=-74.9659011\:\mathrm{a.u.}$$ I don't have Cook's book on hand, so I can't look for the error, but I'd suspect some error as you do.


5

You can find plots and further explanations online (as I easily did). This is a homework-type question, yet I do feel that the information request for diatomic molecular spectroscopic constants warrants at least this partial answer. See this writeup for the "Vibronic Absorption Spectrum of Molecular Iodine" for more details and illustrations. From NIST: T$...


4

$\hat{F}\psi_i=\varepsilon_i \psi_i \implies \langle\phi_k|\hat{F}|\psi_i \rangle = \varepsilon_i\langle\phi_k|\psi_i \rangle$. So the Fock matrix $F_{ki}$ can be approximated by the product of orbital energy $\varepsilon_i$ and the overlap matrix $S_{ki}$. The closer these two values are, the higher the "Eigen-ness". Of course $\phi_k$ and $\psi_i$ have ...


4

I hope I am not mistaken about this, but if I'm wrong then I'll learn so no big deal. I believe this is a result of the fact that the electrons do not experience a true $r^{-1}$ potential due to electron-electron repulsion. Namely, we should expect that the electrons will repel each other which will cause the apparent attraction to the nuclei to be weaker ...


4

The Boys function is $F_n(x)$ a special case of the Kummer confluent hypergeometric function $M(a,b,x) = {_1}F_1(a,b,x)$, which can be found in many special function libraries, such as scipy.special. According to equation (9) of this paper, the relationship is $F_n(x) = \frac{{_1}F_1(n+\frac{1}{2},n+\frac{3}{2},-x)}{2n +1}$ If you use scipy you can get ${...


3

It seems increasing the active space helped make the CASSCF calculation converge. I've had success with increasing the number of active orbitals from 8 to 10. It only took an hour or so.


3

Dissoziation is a typical multi-reference problem, where single-reference methods like Coupled-Cluster usually fail. Therefore CASSCF+MRCI would be the better approach here anyway. If you have a crossing Singlet and Quintuplet state, then you should calculate both. Different spin states do not mix, therefore there is no avoided crossing to be expected. The ...


3

The active space is a truncation of the full CI space. Hence, including more virtual orbitals will lower the energy and eventually approach the FCI limit. The larger the active space, the more accurate the results. The smaller active space will give you a good qualitative description. But if your results turn out to be quantitatively wrong, increasing the ...


3

Variational coupled cluster (vCC) First, you can make a variational CC with a non-unitary T operator in the following way. Recall the variational principle (taken from Wikipedia): $$\varepsilon[\Psi] = \frac{\langle\Psi| \hat{H} |\Psi\rangle}{\langle\Psi|\Psi\rangle}$$ The variational principle states that $\varepsilon \geq E_{0}$, where $E_{0}...


3

The subscript $e$ means measurements relating to the minimum of the internuclear separation, i.e. at the bottom of the potential energy. Many textbooks do not use the subscript $e$ at all but the meaning is the same. The $D_0$ by contrast is the dissociation energy measured from the lowest vibrational energy level ($n=0$), some books also quote $D_e$ which ...


2

Originally, I was convinced that the molecular geometry has very limited influence on the virial ratio: only in so far as the effective basis set will change with the geometry as long as nuclei-centered Slater or Gauss (STO or GTO) basis functions are used.1 Hartree-Fock is accurate as full CI (within the given basis set and its postulates and earlier ...


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