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Reading through some condensed matter physics literature I came across the term "two-centre integrals". Could someone explain what is meant by this in general?

CONTEXT: "the overlap matrix and the largest part of the Hamiltonian matrix elements are given by two-centre integrals. We calculate these in Fourier space..."

chapter 5 of J Phys. Condens. Matter 14 2745

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    $\begingroup$ Could you provide context? For example, you could include the direct quote and a link to the paper. $\endgroup$
    – Kyle Kanos
    Dec 15, 2014 at 16:11
  • $\begingroup$ "the overlap matrix and the largest part of the Hamiltonian matrix elements are given by two-centre integrals. We calculate these in Fourier space..." chapter 5 of J Phys. Condens. Matter 14 2745 $\endgroup$
    – lorenzori
    Dec 15, 2014 at 16:13
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    $\begingroup$ Please edit that information into your post. $\endgroup$
    – Kyle Kanos
    Dec 15, 2014 at 16:14
  • $\begingroup$ It sounds like this is in reference to two centers of force or two molecular centers, etc.. If that's the case, it would cover many effects in solid state physics that are related to two-point correlations. $\endgroup$
    – CuriousOne
    Dec 15, 2014 at 16:16
  • $\begingroup$ OH so it means that it is integrating over for example a field that has two centres? this would make sense $\endgroup$
    – lorenzori
    Dec 15, 2014 at 16:19

1 Answer 1

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In quantum chemistry two-center integrals refer to exchange or coulomb integrals involving 1-electron atomic wave functions (orbitals) centered on two different atoms in a molecule. Say the coulomb repulsion between an electron described by an orbital belonging to atom A and another electron described by orbital belonging to atom B is given in a simplified manner by:

$$\int{\phi_\mathbf{A}^*(1)\phi_\mathbf{B}^*(2)\frac{1}{r_{12}}\phi_\mathbf{A}(1)\phi_\mathbf{B}(2)}\, dr_1dr_2.$$

These and other integrals occur in the Fock matrix, after molecular 1-electron functions are expressed as linear combination of 1-electron atomic functions (Roothan equations).

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