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2

It is very important to note here that $R(r)$ is just the radial part of wavefunction. Wavefunction does not describe any observable. The Born interpretation says that the probability density of finding electron between any two points $x_1$ and $x_2$ is given by $$\rho=\psi\psi^*.$$ While the probability is $$P =\int_{x_1}^{x_2}{\psi\psi^* \mathrm{d}x}.$$ (...


1

Molcas developer here, currently implementing symmetric and canonical orthonormalization for the RASSCF module in addition to Gram-Schmidt. As a rule of thumb you should treat those Orthonormalization schemes (ON-schemes) as different tools for different jobs. If there are $n$ linear independent atomic orbitals (AOs), of which $n$ molecular orbitals (MOs) ...


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The Psi4 project recently published a set of Python notebooks and tutorials on a wide variety of methods, called Psi4NumPy- J. Chem. Theory Comput. 2018 14(7) 3504-3511 - All the code is available as open source on Psi4NumPy GitHub: Hartree Fock including RHF vs. UHF, DIIS, etc. Density Functional - including grids Properties Geometry Optimization ...


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Electrons and photons behave similarly regarding diffraction and interference. First consider a wave going through a single slit, as the amplitude of the waveform and its diffracted component interfere, creating a high central peak and weaker side-lobes. This follows Fraunhofer's formula, which applies to sinusoidal waves. When a photon or an electron beam ...


4

The colours you see in the first two pictures are not electron densities, but phases of the orbitals. The colours of a single molecule could matter if you were calculating the interactions between different molecules (in which in-phase or out-of-phase interactions could matter) but they make no difference between the molecule(s) within the same calculation. ...


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How do I get into method development? As was said in the comments, one would join a group that works in this field and learn from them. From up close, there is some derivation from previous results and some trial and error involved. And yes, a PhD-sized investment is typically required. How do I get into method implementation? It depends on how deep you ...


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We are not sure of the complete context. In simple words, the intensity is proportional to the number of photons impinging on the detector per unit area. This also implies that we can measure the power (in Watts) of the radiation, which is proportional to the number of photons. One can say that a laser is a high intensity source because a lot of photons ...


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A density matrix $D$ is positive semidefinite, hermitian, and has trace one. Because of hermiticity we may assume that it is diagonal. Let's denote the eigenvalues with $\lambda_i$ . Because it is positive semidefinite, we have $0 \leq \lambda_i \leq 1$. The matrix $D^2$ has eigenvalues $\lambda_i^2$. Since it is idempotent ($D = D^2$) and has trace one, we ...


0

The answer "The number of photons of same frequency or wavelength ABSORBED or EMITTED" from your book is a general way of saying when spectral lines are brighter (or darker). Your understanding of this general mechanism is (I think) mostly correct. From what I understand, the differences in intensity of different spectral lines, however, is related to two (...


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As Ivan commented, there are actually an infinite number of possible series of this type. So your question is really why there are only six named series. The reason is part of the culture of science. Typically, a result is named if it is, to use Mithoron's term, sufficiently notable. Thus we have the Diels-Alder reaction, the Hartree–Fock method, the ...


4

Normal modes are technically orthogonal so no energy could flow from one to another, but of course this is just a mathematical construct to make life simple for us, and in reality 'anharmonicity' (physical and electronic) will allow the energy to move about. This is to say that these modes are able to couple to one another. Fermi golden rule is the usual ...


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TL;DR The $\mathrm{d}_{z^2}$ orbital is a result of solving the Schrödinger equation for the hydrogen atom in the most mathematically convenient way. To properly understand this, it is necessary to go back to the fundamentals. The complex d-orbitals are obtained by solution of the Schrödinger equation. In general, these d-orbitals are made up of a radial ...


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It’s just the naming scheme for the hydrogen atom $l=2$ wavefunctions in real form. When you solve the Schrodinger equation you do it in spherical coordinates and the solution you get out involve complex numbers. Thanks to the properties of the Schrodinger equation you can transform these 5 complex valued orbitals in spherical coordinates into 5 new real ...


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