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Resize Project 12 eqns
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‎Project#12/README.md

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@@ -18,27 +18,27 @@ The fundamental idea behind CIS is the representation of the excited-state wave
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functions as linear combinations of singly excited determinants relative to the
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Hartree-Fock reference wave function, *viz.*
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<img src="./figures/singly-excited-determinant.png" height="60">
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<img src="./figures/singly-excited-determinant.png" height="50">
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where *m* identifies the various excited states, and we will use *i* and *j*
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(*a* and *b*) to denote occupied (unoccupied) spin-orbitals. Inserting this
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into the Schr<html>&ouml;</html>dinger equation and left-projecting onto a
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particular singly excited determinant gives
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<img src="./figures/excited-det-schrod-eqn.png" height="60">
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<img src="./figures/excited-det-schrod-eqn.png" height="50">
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If we recognize that we have one of these equations for every combination of
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*i* and *a* spin-orbitals, then this equation may be viewed as a matrix
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eigenvalue problem:
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<img src="./figures/matrix-eigenvalue-problem.png" height="60">
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<img src="./figures/matrix-eigenvalue-problem.png" height="30">
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To solve this equation, we need an expression for the matrix elements in terms
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of things we already know, i.e. Fock matrix elements and two-electron
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integrals. This can be done using either algebraic or diagrammatic techniques
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to obtain (in the spin-orbital notation of previous projects):
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<img src="./figures/matrix-elements.png" height="60">
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<img src="./figures/matrix-elements.png" height="30">
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Our task is then relatively simple: Build the Hamiltonian matrix (expressed in
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the basis of all singly excited determinants) using the above expression and
@@ -72,7 +72,7 @@ from a simple two-electron/two-orbital example (such as the *1s 2s* excited
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state configuration of the He atom). One can easily show that the four
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possible determinants arising from this configuration,
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<img src="./figures/four-possible-determinants.png" height="60">
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<img src="./figures/four-possible-determinants.png" height="50">
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are components of one singlet and one triplet in the following combinations:
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@@ -92,7 +92,7 @@ expression and the equation for the CIS Hamiltonian matrix elements in the
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previous section, we may write a spin-factored equation for the
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<html>&alpha;</html> coefficients as
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<img src="./figures/pin-factored-eqn.png" height="60">
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<img src="./figures/spin-factored-eqn.png" height="30">
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Note that the mix-spin cases (where *j=*<html>&alpha;</html> and
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*b=*<html>&beta;</html> or *vice versa*) do not contribute since the Fock
@@ -101,11 +101,11 @@ carry out spin integration on the integrals in the above expression and assume
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that the <html>&alpha;</html> and <html>&beta;</html> CI coefficients are
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identical for the same spatial orbitals, i.e.,
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<img src="./figures/identical-ci-coeff.png" height="60">
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<img src="./figures/identical-ci-coeff.png" height="30">
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we obtain the <b><i>spatial orbital</i></b> expression
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<img src="./figures/spatial-orbital-expression.png" height="60">
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<img src="./figures/spatial-orbital-expression.png" height="50">
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The part in brackets above is an expression for the spatial-orbital CIS
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Hamiltonian, spin-adapted for singlet excited states, and diagonalization of
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How about the triplets? We use exactly the same spin-factorization, but
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instead require
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<img src="./figures/inverse-ci-coeff.png" height="60">
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<img src="./figures/inverse-ci-coeff.png" height="30">
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This yields a slightly simpler Hamiltonian:
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<img src="./figures/simpler-hamiltonian.png" height="60">
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<img src="./figures/simpler-hamiltonian.png" height="50">
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which, upon diagonalization, will yield only the triplet eigenvalues (but each only occurring once) from your earlier diagonalziation.
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@@ -148,16 +148,16 @@ TDHF/RPA wave function expansion in terms of orbital rotations instead of
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Slater determinants, but that's a discussion for another day.) The TDHF/RPA
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eigenvalue equations take the form
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<img src="./figures/tdhf-eqn.png" height="60">
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<img src="./figures/tdhf-eqn.png" height="50">
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The definition of the ***A*** matrix is just the CIS matrix itself, *viz.*
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<img src="./figures/A-matrix.png" height="60">
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<img src="./figures/A-matrix.png" height="30">
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while **X** and **Y** are the parameters of single excitations and
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de-excitations, respectively, and the ***B*** matrix is simply
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<img src="./figures/B-matrix.png" height="60">
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<img src="./figures/B-matrix.png" height="30">
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Thus, the row/column dimension of the TDHF/RPA Hamiltonian is twice that of the
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CIS Hamiltonian, and the matrix is non-symmetric (so you must be careful about
@@ -174,28 +174,28 @@ Hamiltonian storage cost), one can rearrange the eigenvalue equations. First
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write eigenvalue equation two separate equations, each in terms of the
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submatrices **A** and **B**:
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<img src="./figures/smarter-tdhf-1.png" height="60">
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<img src="./figures/smarter-tdhf-1.png" height="30">
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and
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<img src="./figures/smarter-tdhf-2.png" height="60">
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<img src="./figures/smarter-tdhf-2.png" height="30">
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Now take +/- combinations of these equations to obtain
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<img src="./figures/smarter-tdhf-3.png" height="60">
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<img src="./figures/smarter-tdhf-3.png" height="30">
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and
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<img src="./figures/smarter-tdhf-4.png" height="60">
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<img src="./figures/smarter-tdhf-4.png" height="30">
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Solve for ***(X+Y)*** in the second equation:
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<img src="./figures/smarter-tdhf-5.png" height="60">
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<img src="./figures/smarter-tdhf-5.png" height="30">
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Insert this result into the first equation, rearrange a bit, and finally
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obtain:
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<img src="./figures/smarter-tdhf-6.png" height="60">
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<img src="./figures/smarter-tdhf-6.png" height="30">
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This is an eigenvalue equation of the same dimension as the CIS eigenvalue
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equation (number of occupied orbitals times number of unoccupied orbitals),
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## Test Cases
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The input structures, integrals, etc. for these examples may be found in the
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[input directory](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/input).
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[input directory](./input).
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| Test Case | CIS | RPA (Method 1) | RPA (Method 2) |
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|-----------|-----|----------------|----------------|
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| STO-3G Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_rpa2.txt) |
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| DZ Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_rpa2.txt) |
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| DZP Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_rpa2.txt) |
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| STO-3G Methane | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_rpa2.txt) |
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| STO-3G Water | [output](./output/h2o/STO-3G/output_cis.txt) | [output](./output/h2o/STO-3G/output_rpa1.txt) | [output](./output/h2o/STO-3G/output_rpa2.txt) |
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| DZ Water | [output](./output/h2o/DZ/output_cis.txt) | [output](./output/h2o/DZ/output_rpa1.txt) | [output](./output/h2o/DZ/output_rpa2.txt) |
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| DZP Water | [output](./output/h2o/DZP/output_cis.txt) | [output](./output/h2o/DZP/output_rpa1.txt) | [output](./output/h2o/DZP/output_rpa2.txt) |
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| STO-3G Methane | [output](./output/ch4/STO-3G/output_cis.txt) | [output](./output/ch4/STO-3G/output_rpa1.txt) | [output](./output/ch4/STO-3G/output_rpa2.txt) |
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