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This is a very broad question, so I am going to give a very brief overview of typical exponentially-scaling problems. I am not an expert in most of these areas, so any suggestions or improvements will be welcome. Solving the Schrödinger equation In order to solve the Schrödinger equation numerically, you need to diagonalise a rank $3N$ tensor -- as you can ...


20

ORCA The local pair-natural orbital (DLPNO) based coupled cluster method have been managed to investigate large organic molecule and small proteins (linear C150H302 (452 atoms, >8800 basis functions) , Crambin with 644 atoms, and more than 6400 basis functions ,C350H902 (>1000 atoms, > 20000 basis functions)). Riplinger, C., Sandhoefer, B., Hansen, ...


19

This is not going to be a full answer, but maybe a starting point, a pointer to somewhat troubled systems. The general keyword here is degeneracy. Whenever you have fewer electrons than fit in the degenerate orbitals, two things may happen: In the simple case the molecule will distort, see aromaticity and anti-aromaticity. In the complicated case, you ...


18

The inclusion of spin-orbit coupling in electronic structure calculations is done by including the interaction between the electron-spin and the orbital angular momentum in the Hamiltonian. Such interaction is described according to the spin-orbit Hamiltonian defined as follows, $$ \hat{H}_\mathrm{SO} = \frac{1}{2m_\mathrm{e}\mathrm{c}^2} \frac{1}{r} \left( ...


17

Geometry optimization corresponds to a system in equilibrium. It is the "average" position of a molecule vibrating in a well. However, there are many cases where the system is "non-equilibrium" which are important to model. For example, photochemistry can involve important non-equilibrium processes. In photochemistry, relaxation to the ground-state ...


17

Aligning a molecule to a particular frame of reference (e.g., with the z-axis along a particular bond) is part of Avogadro for this reason: In Avogadro 1.x, there's an align tool You click one atom that will be set to the origin You click another atom that will be projected along the x, y, or z-axis You click 'align' and the molecular coordinates will be ...


16

Quantum computers provide the possibility of simulating systems that are so mathematically complex that classical computers cannot practically be used. The difficulty comes from the fact that the electrical properties of materials, and other chemical systems, are governed by the laws of many body quantum mechanics, which contain equations that are extremely ...


16

Basically there are two kinds of approaches which may be found in many text books, L-S coupling and j-j coupling. L-S coupling means that scalar electronic states (e.g. atomic L-S states and linear molecular Lambda-S states) are calculated first, and then the SO matrix is constructed with the help of 1-e (and optional 2-e) SO integrals. After diagonalizaion,...


16

There is linear scaling DFT, where you use density matrix as your object throughout SCF iterations instead of the wavefunctions. The linear scaling comes from the sparsity assumption/truncation on the density matrix. Usually, the wavefunctions are dense vectors, and any eigensolver results in cubic scaling with respect to the number of electrons/bases. ...


15

A rigorous definition of a semimetal can for example be found in the mathematical literature (e.g. this paper: arXiv:2002.01990v1 [math-ph], pdf version, assumption 2.5). The basic idea is that the Fermi surface, i.e. the union of points where occupied an unoccupied bands meet, consists exactly of isolated points, the Dirac points. These points are what is ...


15

It is hard to claim that any FCI code overcomes the exponential wall, especially for strongly correlated systems. There are many algorithms, e.g. CDFCI, HCI, FCIQMC, ACI, etc., which significantly reduce the computational cost of direct FCI calculations and represent wavefunctions by sparse vectors. However, in my opinion, all of them only reduce the ...


15

As Martin has said a practical way to decide if a molecule requires multi-reference treatment is to calculate the unrestricted Hartree Fock natural orbitals.[1] Significant fractional occupancy of a UHF natural orbital indicates that the orbital should be included in the active space. For a nonorthogonal finite basis set, they are given as $$\mathbf{S}(\...


15

Even for the simplest transition metal diatomic molecules, the most reliable way to know the ground-state spin configuration is often by doing experiments. I will give an example where it's easy to correctly determine the ground state spin configuration, and then an example where it has remained impossible as of the year 2020. Cr$_2$: Here we can accurately ...


14

When to include relativistic corrections or modeling of any kind in computational methods is a rather complex one. Full Dirac methods as you asked about (DHF) recapture two important factors, so called scalar relativistic effects, and spin effects. I'll elaborate on each and when including them is important. Scalar Relativistic Effects - This largely ...


14

The question is too broad to be answered directly so I will provide a somewhat general scheme. Basically in an integral like $$ \int d\mu A B C $$ one would seek to expand each part in irreducible representations of a given group, say for instance \begin{align} B=\frac{1}{\vert \mathbf{r}-\mathbf{r}^\prime\vert} =\frac{1}{r} \sum_{\ell} \left(\frac{r'}{r}\...


13

Quantum computing has been extensively discussed as long as the first quantum computers have become a reality. As already pointed out, we are still at very early stages of such development. What we need to keep in mind is that a quantum computer needs a quantum computing algorithm. The latter can have few to no correlation with classical algorithms, such as ...


13

Multiwfn The software Multiwfn can calculated four aromaticity indices, namely FLU, FLU-pi, PDI and information-theoretic index. Multiwfn can perform wavefunction analyses based on outputted file of almost all well-known quantum chemistry programs, such as Gaussian, ORCA, GAMESS-US, Molpro, NWChem, Dalton, xtb, PSI4, Molcas, Q-Chem, MRCC, deMon2k, Firefly, ...


12

Introduction Your question reminds me of a quote by Paul Dirac, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes ...


12

The curse of dimensionality is indeed a huge problem in quantum chemistry, since the possible ways N electrons can occupy K orbitals is a binning problem whose computational cost grows factorially (almost as fast as x^x!) with the size of the system. Moreover, for accurate results you need K>>N in order to account for the so-called dynamical correlation, ...


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

Your supervisor is correct that in almost all practical cases in quantum chemistry, the cost savings of using Gaussian-type orbitals far outweigh the disadvantages of needing more orbitals. First of all, there is a bit of a misconception that Slater-type orbitals are far more accurate. The motivation for using Slater-type orbitals is due to their resemblance ...


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 ...


11

Quantum computers potentially could provide a far more efficient way to solve the electronic structure problem: finding the ground state energy (or other properties) of electrons in a compound. The most prominent quantum algorithms for this purpose are the Phase Estimation Algorithm (PEA) and the Variational Quantum Eigensolver (VQE). While PEA offers the ...


11

Creating amorphous structures is non-trivial and always rather expensive. Generally you are using large supercells such that you can achieve mid-range disorder. That means, since the cell is still periodic, you cannot truly achieve the long range disorder of amorphous materials, but in a medium range (> the unit cell volume) you can. As such, we some times ...


11

Is this calculated indirect bandgap at room temp. or at 0K? QE is based on the density functional theory (DFT). DFT is a ground-state (0K) theory and hence the calculated bandstructure is 0K. If it is at room temperature the indirect bandgap should be around 1.1 eV as from the literature. So is there any energy correction factor that has to be performed to ...


10

I’ll expand on this later but here is the abbreviated version. CTF The CTF code can do very large iterative CCSD and CCSDT using a dense spin-orbital formalism. CCSD up to 55 (50) water molecules with cc-pVDZ http://solomonik.cs.illinois.edu/talks/molssi-monterey-may-2017.pdf The 8-water CCSDT problem in Table 3 took 15 minutes on 1024 nodes of BG/Q and ...


10

I will list 5 issues with F12/R12 methods, and then try to explain them the best I can: Need for an auxiliary basis set Most (if not all) F12/R12 methods require more than just a standard single-particle Gaussian basis set. For one of the most common elements (carbon) in electron structure calculations, you can see some of these auxiliary basis sets on Basis ...


10

Spherical harmonics are not themselves full atomic orbitals. Consider the Hydrogen wave function, which separates into a radial part and an angular part. The latter is a spherical harmonic, but the former is some other function (in the case of Hydrogen it's a Laguerre polynomial). In general, we can approximate the angular part for other atoms with the same ...


10

The empty orbitals are always included in plane wave calculations. The plane waves don't conform to atomic symmetry, and can be seen to contain contributions of arbitrary angular momentum. If you were to use an atomic basis set, then you would need to put in D basis functions explicitly; however, this is pretty much always done in practice at least in the ...


10

I would like to add a few clarifications to Jack's answer: Standard DFT calculations with fixed ionic positions are actually not even 0 K. A better way to describe them is that they are static lattice calculations. The difference between static lattice and 0 K is the contribution from quantum zero-point fluctuations. This contribution is generally small, ...


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