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Quantum Chemistry Methods

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Luminescent Materials

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 322))

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Abstract

The methods of Quantum Chemistry used in this book to study crowded manifolds of local excited states of luminescent materials are introduced in this chapter. The embedded-cluster approximation for local states is discussed first, which allows to treat them with the same wave function theory tools used in quantum chemistry for isolated molecules. Then, the chapter goes through the two-component Douglas-Kroll-Hess relativistic Hamiltonian derived from the four-component Dirac Hamiltonian and analyzes its scalar and spin-orbit coupling components. The last part is dedicated to the discussion of the very important but difficult to handle electron correlation and its forms: The static correlation responsible for the configurational multiplets and its handling with CASSCF and RASSCF multiconfigurational variational methods. The dynamic correlation necessary for quantitative accuracy and its handling with MS-CASPT2 and MS-RASPT2 multi-reference perturbational methods. And the simultaneous consideration of electron correlation and spin-orbit coupling by means of the RASSI-SO method.

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Notes

  1. 1.

    The Dirac (kinetic energy) Hamiltonian for a free electron \(\underline{\hat{h}}_\mathrm{D}\) is a 4\(\,\times \,\)4 matrix of operators and the one-electron wave function \(\underline{\phi }\) (or spinor) is a 4\(\,\times \,\)1 vector of 1-electron functions, so that the Dirac equation for one free electron of energy \(\varepsilon \) reads \(\underline{\hat{h}}_\mathrm{D}\,\underline{\phi } = \varepsilon \underline{\phi }\). This equation is equally valid if operator and wave function undergo 4\(\,\times \,\)4 unitary transformations (with an arbitrary 4\(\,\times \,\)4 matrix \(\underline{u}\) that fulfils \(\underline{u}^\dagger \,\underline{u}=\underline{u}\,\underline{u}^\dagger =\boldsymbol{1}\)). In effect, \(\underline{u}\,\underline{\hat{h}}_\mathrm{D}\,\underline{u}^\dagger \underline{u}\underline{\phi } = \varepsilon \underline{u}\,\underline{\phi }\), which gives the equivalent, transformed Dirac equation \(\underline{\hat{h}}_\mathrm{D}^\prime \underline{\phi ^\prime } = \varepsilon \underline{\phi ^\prime }\), with the transformed Hamiltonian \(\underline{\hat{h}}_\mathrm{D}^\prime = \underline{u}\,\underline{\hat{h}}_\mathrm{D}\,\underline{u}^\dagger \) and spinor \(\underline{\phi ^\prime } = \underline{u}\underline{\phi }\).

  2. 2.

    In an \(N\)-electron system, the probability of finding an electron (any of them) in a differential volume dV around a point (x, y, z) is \(\gamma (x,y,z)dV\), where \(\gamma \) is known as the one-particle density function, or electron probability density, or electron density, which varies from point to point and has units of \(V^{-1}\) (as corresponds to being a density). Equivalently, the probability of simultaneously finding an electron in a differential volume \(dV_1\) around a point \((x_1,y_1,z_1)\) and another in a differential volume \(dV_2\) around a point \((x_2,y_2,z_2)\) is \(\Gamma (x_1,y_1,z_1,x_2,y_2,z_2)dV_1dV_2\), where \(\Gamma \) is known as the two-electron probability density or two-electron density function, which depends on the pair of points and has units of \(V^{-2}\) (as corresponds to being a 2-particle density). The one- and two-electron density functions \(\gamma \) and \(\Gamma \) of an \(N\)-electron system are calculated from its many-electron wave function \(\Psi \) (see below), which is a function of the position coordinates x, y, z and spin coordinates \(\sigma \) of all the \(N\) electrons.

    When \(\Gamma (x_1,y_1,z_1,x_2,y_2,z_2) = \gamma (x_1,y_1,z_1) \gamma (x_2,y_2,z_2)\), the wave function \(\Psi \) is said to be uncorrelated, because the probability of finding an electron at point 1 when another electron is already at point 2 (which is calculated substituting \(\gamma (x_2,y_2,z_2)\) by 1 in the formula for \(\Gamma \)) is independent of where the second electron is, regardless of its distance from point 1.

    Correspondingly, when \(\Gamma (x_1,y_1,z_1,x_2,y_2,z_2) \ne \gamma (x_1,y_1,z_1) \gamma (x_2,y_2,z_2)\), i.e. when \(\Gamma (x_1,y_1,z_1,x_2,y_2,z_2) = \gamma (x_1,y_1,z_1) \gamma (x_2,y_2,z_2) [1+f(x_1,y_1,z_1,x_2,y_2,z_2)]\), the probability of finding an electron at point 1 when another electron is at point 2 depends on the position of the second electron (through the function f) and the wave function \(\Psi \) is said to be correlated. The function \(f(x_1,y_1,z_1,x_2,y_2,z_2)\) is a measure of correlation.

    \(\gamma \) is calculated by integrating \(\Psi ^\star \Psi \) over the spatial x, y, z and spin \(\sigma \) coordinates of all \(N\) electrons but one, plus the spin coordinate of the remaining electron, and a factor \(N\) due to the fact that all electrons have the same probability of being at a given point. I.e., \(\gamma (x_1,y_1,z_1) = N\int _{(x_i,y_i,z_i,\sigma _i)_{i=2,\ldots ,N}} \int _{\sigma _1} \Psi ^\star \Psi d\sigma _1 d(x_i,y_i,z_i,\sigma _i)_{i=2,\ldots ,N}\). \(\Gamma \) is calculated the same way, but integrating the coordinates of all electrons but two: \(\Gamma (x_1,y_1,z_1,x_2,y_2,z_2) = N(N-1) \int _{(x_i,y_i,z_i,\sigma _i)_{i=3,\ldots ,N}} \int _{\sigma _1}\int _{\sigma _2} \Psi ^\star \Psi d\sigma _1d\sigma _2 d(x_i,y_i,z_i,\sigma _i)_{i=3,\ldots ,N}\).

  3. 3.

    A spin-orbital \(\psi \) is a one-electron function that results from the product of an orbital \(\varphi \) and a spin function \(\alpha \) or \(\beta \): \(\psi = \varphi \alpha \) or \(\psi = \varphi \beta \).

  4. 4.

    We must be aware of a potentially confusing notation here: Although all multiconfigurational wave functions of the kind (1.43) are in fact CI functions, the name CI methods (and their corresponding CI wave functions) is commonly reserved to the ones now described.

  5. 5.

    In the single and double virtual electron excitations in RASPT2, the electrons go from inactive and active orbitals (i.e. with double and variable occupancy, respectively, in all RASSCF wave functions) to active and virtual (empty) orbitals. Normally, excitations from the innermost inactive orbitals are excluded; in that case, the orbitals are called frozen and dynamic correlation is not considered for their electrons. Choices that allow to exclude dynamic correlation for core electrons while including it for outer-valence and, optionally, inner-valence electrons are commented in Sect. 2.3 and throughout the applications sections.

  6. 6.

    Interestingly, this option is formally a one-step spin-orbit coupling calculation, because the diagonalization of the matrix of \(\hat{H}_\mathrm{EC}^\mathrm{eff}\) on the {\(\Phi ^\text {RAS}_i\)} basis, which is required to obtain the spin-orbit-free MS-CASPT2 energies, is not necessary.

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Authors and Affiliations

  1. Departamento de Química, Universidad Autónoma de Madrid, Madrid, Spain

    Zoila Barandiarán

  2. Department of Solid State Sciences, Ghent University, Gent, Belgium

    Jonas Joos

  3. Departamento de Química, Universidad Autónoma de Madrid, Madrid, Spain

    Luis Seijo

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  1. Zoila Barandiarán
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Barandiarán, Z., Joos, J., Seijo, L. (2022). Quantum Chemistry Methods. In: Luminescent Materials. Springer Series in Materials Science, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-030-94984-6_1

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