Computational Chemistry 4.11 - One-Electron Integrals

Short lecture on one-electron integrals of spin orbital expectation values.
Terms in the total energy of a molecular system are either of the one-electron or two-electron form, depending on the number of electrons in the operator. One-electron terms include electron kinetic energy and electron-nuclear attraction. One-electron terms can be factored to only depend on the coordinates of one-electron at a time, and are thus "separable". Two-electron terms (electron-electron repulsion) are not separable, and thus the primary source of difficulty in obtaining molecular energy expressions. This video performs a derivation to show how the one-electron energy of a molecular energy depends only on the core Hamiltonian operator applied to each occupied spin orbital, multiplied by its complex conjugate, and integrated over all space (i.e. one-electron integrals).
Notes Slide: i.imgur.com/Pt...
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[@r__9_1____a34]

You have the best KZbin channel

[@solsticetwo3476]

In the integral definition of h_i (the last equation) the reference to electron 1 is unnecessary and could cause confusion . It is better to use the dummy variable 'x'. The energy is determined by the orbital since the electrons are "indistinguishable" and each contribute to 1/N to the energy of the orbital.

[@Mehdi-el8dt]

l3zz a3chiri

[@b6kf367]

this comment and the one about dummy index of O^(2) operator clarified so much hair plucking, thanks :)

[@rimjhimshukla365]

1. Why do we call it one-electron energy even though we are multiplying it with 2 in the end to describe two electrons? 2. Please refer to the aqua blue equation in the right column (almost at the end), how do we know that 1 and 2 in braket notation denote spin-orbital and not the electrons. I'm getting a little confused with the notation, would be really glad if you would help me.

[@nimehg5734]

i think; 1) he has taken 2 electron system. By using one electron operator on one of the electron and multiplying it by 2 gives energy of two electrons excluding electron-electron interaction. 2) 1 and 2 are spin-orbitals of electron 1 and 2 respectively which means they represent electron 1 and 2 only. Spin orbitals are wavefunctions based on position and spin of the electron.

[@faysalmumtaz]

Thank you so much for the lectures. They are very useful. I have a small question regarding the last equation of the slide where you write the Hamiltonian h_i in integral form. Why the integral involves only dx_1, h_1 and chi_i(1)? shouldn't this be general i.e. dx_i, h_i and chi_i(I)?

[@TMPChem]

The integral is for electron 1. This is an arbitrary index assigned to the electron. All that matters is that the integral over the coordinates (d(chi)1) matches the index of the electron (electron 1). We're saying the one-particle Hamiltonian matrix element h_ii is equal to the integral over all the coordinates of the complex conjugate of spin orbital i times the operator h_i acting on spin orbital i. The spin orbital is orbital i, some number between 1 and 2K (number of basis functions). The electron is some number between 1 and N, but since all electrons are indistinguishable, it's easiest to just label it as electron 1.

[@faysalmumtaz]

TMP Chem many thanks for the clarification

[@jessesonnenschein5665]

Why do you multiply by two almost at the end? That implies that operator for electron 1 is equal to operator for electron 2...but are they the same so I can multiply by two de last 2 integral summation ? Thank you.

[@chaosdrache09]

You are saying that it is the same procedure for electron 2, but for electron 2 you integrate first over x1. Or can you just rename them? But that would imply that h1 on chi1 = h2 on chi2 and also h1 on chi2 = h2 on chi1 have the same Energyeigenstate.

[@TMPChem]

I think you're on the right track. Electron one's operator acting on electron has the same effect as electron two's operator acting on electron two, and likewise when the labels are mismatched. This is because no electron is privileged relative to any other, as they are all indistinguishable. The labels 1, 2, ... are arbitrary and can just as easily be traded as labels without affecting the outcome.

[@NavneetKumar-oe7hh]

Derive the exchange and coloumb integral for two interacting electron with alpha and beta spin. Please derive sir you r my last hope i have to submit till 12 pm

[@lonelalydmitri9489]

@1:07 , it's fancy i guess

[@vigneshwarankannan4999]

what is the basis set you have chosen...will that be completely integrated out without numerical calculation ?

[@TMPChem]

Hi Vigneshwaran. The math presented here is completely independent of one's choice of basis set, or whether there even is a basis set. All we know for sure is that there is some function which represents each molecular orbital. Most frequently this is chosen to be a linear combination of some number (K) of atomic orbital functions, but the idea is the same whether we do so in terms of atomic orbitals or one complete molecular orbital function. Whatever the representation is, we plug in the function to this formula, and go about calculating the effect of the operator acting on the orbital multiplied by its complex conjugate. When using a basis set, we can break this down further, but for now that is abstracted away.

[@dwaipayanroychowdhury5279]

Can you recommend me some books regarding these math calculation??

[@TMPChem]

The classic introductory graduate text in electronic structure theory is Modern Electronic Structure Theory by Szabo and Ostlund, which details the derivation and components of many of these methods / algorithms, but is still quite scant on details as to the mechanics of how to compute numerical values for these integrals. For that, you'll want to examine the brutal gauntlet that is Molecular Electronic Structure Theory by Helgaker, Jorgensen, and Olsen, often referred to by researchers in the field as "the purple book".

[@dwaipayanroychowdhury5279]

@@TMPChem thanks man..you're the best

[@apburner1]

First.

[@TMPChem]

Perhaps.

[@nimehg5734]

@@TMPChem lol too much free time to reply these kind of comment