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QuantumChemistry

 ExcitonPopulations
 compute the exciton populations

 Calling Sequence ExcitonPopulations(molecule, method = name/procedure, options) ExcitonPopulations(molecule, data, options)
 Parameters

 molecule - list of lists; each list has 4 elements, the string of an atom's symbol and the atom's x, y, and z coordinates method - method = name/procedure where name/procedure is a method that can return a correlated 2-RDM such as data - table; table returned by one of the electronic structure methods listed under method containing a 2-RDM options - (optional) equation(s) of the form option = value where option is one of excitontype, nexcitons, excitonmatrix, showtable, returnvecs or, when called with the method keyword, any option of the method such as basis, charge, spin, symmetry, active, ect.

Description

 • The ExcitonPopulations command computes the occupations (populations) of a molecule's exciton states.
 • Excitons are quasi-particles that consist of an electron and a hole (the absence of an electron).  The exciton populations and states are computed from the eigenvalues and eigenvectors of the particle-hole matrix (G matrix) or its modified form (Gt matrix).  The exciton population can become large on the order of the total number of excitons (electron-hole pairs).  The phenomenon of having a large number of excitons in the same exciton state is known as exciton condensation.  Recently, exciton condensates have been experimentally realized in graphene double layers and van der Waals heterostructures.  Refer to the References for further details.
 • The keyword nexcitons sets the number of exciton states (default = 5) to return.  It can be an integer or the string "all".
 • The keyword excitontype sets the spin of the exciton states to "singlet" (default) or "triplet".
 • The keyword excitonmatrix determines whether the particle-hole matrix (G matrix) or its modified form (Gt matrix) is used.
 • The populations (eigenvalues) are returned by default.  Using the keyword returnvecs = true, we can return the populations as a Vector and the exciton states (eigenvectors) as a Matrix.
 • The keyword showtable = true returns the exciton populations in a fancy table in the Maple worksheet.
 Options
 • nexcitons = nonnegint/string -- an integer or the string "all" determines the number of exciton states to return.  Default is 5.
 • excitontype = string -- "singlet" or "triplet" sets the spin of the computed exciton states.   Default is "singlet".
 • excitonmatrix = string -- "G" or "Gt" for the G-matrix or the modfied G-matrix.   Default is "Gt".
 • showtable = truefalse -- when true, a fancy table is printed to the worksheet. Default is false.
 • returnvecs = truefalse -- when true, the exciton states are returned as columns of a Matrix.  Default is false.

References

 1 S. Safaei and D. A. Mazziotti, Phys. Rev. B 98, 045122 (2018). "Quantum signature of exciton condensation"
 2 Anna O. Schouten, LeeAnn M. Sager, and David A. Mazziotti, J. Phys. Chem. Lett. 12 9906 (2021). "Exciton Condensation in Molecular-Scale van der Waals Stacks"
 3 X. Liu  and J. I. A. Li, K. Watanabe, T. Taniguchi, J. Hone, B. I. Halperin, P. Kim, and C. R. Dean, Science 375, 6577 (2022). "Crossover between strongly coupled and weakly coupled exciton superfluids"

Examples

 > $\mathrm{with}\left(\mathrm{QuantumChemistry}\right):$

Consider the benzene molecule

 >
 ${\mathrm{benzene}}{≔}\left[\left[{"C"}{,}{-1.21310000}{,}{-0.68840000}{,}{0}\right]{,}\left[{"C"}{,}{-1.20280000}{,}{0.70640000}{,}{0.00010000}\right]{,}\left[{"C"}{,}{-0.01030000}{,}{-1.39480000}{,}{0}\right]{,}\left[{"C"}{,}{0.01040000}{,}{1.39480000}{,}{-0.00010000}\right]{,}\left[{"C"}{,}{1.20280000}{,}{-0.70630000}{,}{0}\right]{,}\left[{"C"}{,}{1.21310000}{,}{0.68840000}{,}{0}\right]{,}\left[{"H"}{,}{-2.15770000}{,}{-1.22440000}{,}{0}\right]{,}\left[{"H"}{,}{-2.13930000}{,}{1.25640000}{,}{0.00010000}\right]{,}\left[{"H"}{,}{-0.01840000}{,}{-2.48090000}{,}{-0.00010000}\right]{,}\left[{"H"}{,}{0.01840000}{,}{2.48080000}{,}{0}\right]{,}\left[{"H"}{,}{2.13940000}{,}{-1.25630000}{,}{0.00010000}\right]{,}\left[{"H"}{,}{2.15770000}{,}{1.22450000}{,}{0}\right]\right]$ (1)

We perform a variational 2-RDM calculation with a 6-electrons-in-6-orbitals [6,6] active space

 >

After the calculation we use the command ExcitonPopulations to compute the exciton populations, the number of excitons in a given exciton (particle-hole) state

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 Exciton State Exciton Population $1$ $1.00564746$ $2$ $1.00559656$ $3$ $0.99287953$ $4$ $0.95440962$ $5$ $0.95438149$ $6$ $0.95385218$ $7$ $0.95361622$ $8$ $0.95208482$

 >