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Rayleigh theorem for eigenvalues

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The Rayleigh Theorem for Eigenvalues - The Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh, Lord Rayleigh, and 3rd Baron Rayleigh are the titles of John William Strutt, after the death of his father, the 2nd Baron Rayleigh. Lord Rayleigh made extensive contributions, not just to both theoretical and experimental physics, but also to applied mathematics. The Rayleigh theorem for eigenvalues is a bedrock of self-consistent, variational calculations of electronic and related properties of materials, from atoms, molecules, and nanostructures to semiconductors, insulators, and metals. Except for metals, most of these other materials have an energy or a band gap, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals, the energy spectrum is in bands and we speak of a band gap, if any, as opposed to energy gap. Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including Parseval's theorem. For this reason, keeping the full name of "Rayleigh Theorem for Eigenvalues" avoids confusions.

STATEMENT OF THE THEOREM

The theorem, as indicated above, applies to the resolution of equations called eigenvalue equations. i.e., the ones of the form HѰ = λѰ, where H is an operator, Ѱ is a function and λ is number called the eigenvalue. To solve problems of this type, we expand the unknown function Ѱ in terms of known functions that constitute a complete set. The expansion coefficients are also number. The number of known functions included in the expansion, the same as that of coefficients, is the Hamiltonian dimension of the matrix that will be generated. The statement of the theorem follows. [1][2]

Let an eigenvalue equation be solved by linearly expanding the unknown functions in terms of N known functions. Let the resulting eigenvalues be ordered from the smallest (lowest), λ1, to the largest (highest), λN. Let the same eigenvalue equation be solved using a basis set of dimension (N+1) that comprises the previous N functions plus an additional one. Let the resulting eigenvalues be ordered from the smallest , λ'1, to the largest, λ'(N+1). Then, the Rayleigh theorem for eigenvalues states that λ'i ≤ λi for i = 1 to N.

A subtle point about the above statement is that the smaller of the two sets of functions must be a subset of the larger one. The above inequality does not hold otherwise.

SELF CONSISTENT CALCULATIONS

In quantum mechanics,[3] where the operator H is the Hamiltonian, the eigenvalues that are occupied (by electrons) up to the applicable number of electrons; the remaining eigenvalues, not occupied by electrons, are empty energy levels. The energy content of the Hamiltonian is trivially the sum of the occupied eigenvalues. The Rayleigh theorem for eigenvalues is extensively utilized in calculations of electronic and related properties of materials. Specifically, the electronic energies of materials are obtained through calculations said to be self-consistent, as explained below.

In density functional theory (DFT) calculation of electronic energies of materials, the eigenvalue equation, HѰ = λѰ, has a companion equation that gives the electronic charge density of the material in terms of the wave functions of the occupied energies. To be reliable, these calculations have to be self-consistent, as explained below.

The process of obtaining the electronic energies of a material begins with the selection of an initial set of known functions (and related coefficients) in terms of which one expands the unknown function Ѱ. Using the known functions for the occupied states, one constructs an initial charge density for the material. For density functional theory calculations, once the charge density is known, the potential, the Hamiltonian, and the eigenvalue equation are generated. Solving this equation leads to eigenvalues (occupied or unoccupied) and their corresponding wave functions (in terms of the known functions and new coefficients of expansion). Using only the new wave functions of the occupied energies, one repeat the cycle of constructing the charge density, generating the potential and the Hamiltonian. Then, using all the new wave functions (for occupied and empty states), one regenerates the eigenvalue equation and solves it. Each one of these cycles is called an iteration. The calculations will be complete when the difference between the potentials generated in Iteration (N+1) and the one immediately preceding it (i.e., n) is 10-5 or less. The iterations are said to have converged and the outcomes of the last iteration are the self-consistent results that are supposed to describe the material.

APPLICATION OF THE RAYLEIGH THEOREM FOR EIGENVALUES IN DENSITY FUNCTIONAL THEORY (DFT)

The characteristics and number[1][2] of the known functions utilized in the expansion of Ѱ naturally has a bearing on the quality of the final, self-consistent results. The selection of atomic orbitals that include exponential or Gaussian functions, in additional to polynomial and angular features that apply, practically ensure the high quality of self-consistent results, except for the effects of the size[1][2] and of attendant characteristics of the basis set. These characteristics include the polynomial and angular functions that are inherent to the description of s, p, d, and f states for an atom. While the s functions[4] are spherically symmetric, the others are not; they are often called polarization orbitals or functions.

The issue is the following. Density functional theory is for the description of the ground state of materials, i.e., the state of lowest energy. The second theorem[5] [6] of DFT states that the energy functional for the Hamiltonian [i.e., the energy content of the Hamiltonian] reaches its lowest value (i.e., the ground state) if the charge density employed in the calculation is that of the ground state. We described above the selection of an initial basis set in order to perform self-consistent calculations. A priori, there is no known mechanism for selecting a basis set so that , after self consistency, the charge density it generates is that of the ground state. Self consistency with a given basis set leads to the lowest energy content of the Hamiltonian for that basis set. As per the Rayleigh theorem for eigenvalues, upon augmenting that initial basis set, the ensuing self consistent calculations lead to an energy content of the Hamiltonian that is lower than or equal to that obtained with the initial basis set. Indeed, as per the Rayleigh theorem for eigenvalues, the lowest energy obtained with a basis set, after self consistency, is relative to that basis set. A larger basis set that contains the first one could lead to a lower, self consistent energy. One may paraphrase the issue as follows. Several basis sets of different sizes, upon the attainment of self-consistency, lead to stationary (converged) solutions. There exists an infinite number of such solutions.. A priori, one has no means to find out the particular basis set which, after self consistency, leads to the ground state charge density of the material, and, according to the second DFT theorem, to the ground state energy of the material under study.

The solution of the problem follows. Let us first recall that self-consistent density functional theory calculations, with a single basi set, produce a stationary solution which cannot be claimed to be that of the ground state. To find the DFT ground state of a material, one has to vary[5] [6] the basis set (in size and others) in order to minimize the energy content of the Hamiltonian, while keeping the number of particles constant. Hohenberg and Kohn,[5] specifically stated that the energy content of the Hamiltonian "has a minimum at the ‘correct’ ground state Ψ, relative to arbitrary variations of Ψ′ in which the total number of particles is kept constant.” Hence, the trial basis set is to be varied in order to minimize the energy. The Rayleigh theorem for eigenvalues shows how to perform such a minimization.. The first trial basis set has to be a small one that account for all the electrons in the system. One ten augment this basis set with one atomic orbital . Depending on the s, p, d, or f character of this orbital, the size of the new basis set (and the dimension of the Hamiltonian matrix) will be larger than that of the initial one by 2, 5, 10, of 14, respectively. Given that the initial, trial basis set was deliberately selected to be small, the resulting self consistent results cannot be assumed to describe the ground state of the material. Upon performing self-consistent calculations with the augmented basis set, one compares the occupied energies from CalculationsI and II, after setting the Fermi level to zero. Invariably,[7] [8] the occupied energies from Calculation II and lower than or equal to their corresponding values from Calculation I. Naturally, one cannot affirm that the results from Calculation II describe the ground state of the materials, given the absence of any proof that the occupied energies can be lowered further. Hence, one continues the process of augmenting the basis set with one orbital and of performing the next self-consistent calculation. The process is complete when three consecutive calculations yield the same occupied energies. One can affirm that the occupied energies from these three calculations represent the ground state of the material. Indeed, with while two consecutive calculations can produce the same occupied energies, these energies may be for a local minimum as opposed to the absolute minima of the occupied energies. To have three consecutive calculations produce the same occupied energies is the robust criterion[9][10] for the attainment of the ground state of a material (i.e., the state where the occupied energies have their absolute minimal values).

Even though the paragraph above shows how the Rayleigh theorem enables the generalized minimization of the energy content of the Hamiltonian to reach the ground state, we are left with the fact that three different calculations produce This ground state. While their the occupied energies are the same, the unoccupied energies are not identical. Indeed, the general trend is that the unoccupied energies from the calculations are in the reverse order of the sizes of the basis sets for these calculations.






BIBLIOGRAPHY

  1. ^ a b c Gould, S. H. (1966-12-31). Variational Methods for Eigenvalue Problems. Toronto: University of Toronto Press. ISBN 978-1-4875-9600-2.
  2. ^ a b c Sähn, S. (1971). "A. D. Kovalenko, Thermoelasticity. 251 S. m. Fig. Groningen 1969. Wolters-Noordhoff Publishing. Preis S 11.00". ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik. 51 (1): 72–72. doi:10.1002/zamm.19710510132. ISSN 0044-2267.
  3. ^ CALLAWAY, J. (1974). Quantum Theory of the Solid State (Student Edition). OCLC 986331165.
  4. ^ Harmon, B. N.; Weber, W.; Hamann, D. R. (1982-01-15). "Total-energy calculations for Si with a first-principles linear-combination-of-atomic-orbitals method". Physical Review B. 25 (2): 1109–1115. doi:10.1103/physrevb.25.1109. ISSN 0163-1829.
  5. ^ a b c Hohenberg, P.; Kohn, W. (1964-11-09). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864 – B871. doi:10.1103/physrev.136.b864. ISSN 0031-899X.
  6. ^ a b Kohn, W.; Sham, L. J. (1965-11-15). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133 – A1138. doi:10.1103/physrev.140.a1133. ISSN 0031-899X.
  7. ^ Zhao, G. L.; Bagayoko, D.; Williams, T. D. (1999-07-15). "Local-density-approximation prediction of electronic properties of GaN, Si, C, andRuO2". Physical Review B. 60 (3): 1563–1572. doi:10.1103/physrevb.60.1563. ISSN 0163-1829.
  8. ^ Bagayoko, Diola (2014-12). "Understanding density functional theory (DFT) and completing it in practice". AIP Advances. 4 (12): 127104. doi:10.1063/1.4903408. ISSN 2158-3226. {{cite journal}}: Check date values in: |date= (help)
  9. ^ Ekuma, C.E.; Jarrell, M.; Moreno, J.; Bagayoko, D. (2013-11). "Re-examining the electronic structure of germanium: A first-principle study". Physics Letters A. 377 (34–36): 2172–2176. doi:10.1016/j.physleta.2013.05.043. ISSN 0375-9601. {{cite journal}}: Check date values in: |date= (help)
  10. ^ Franklin, L.; Ekuma, C.E.; Zhao, G.L.; Bagayoko, D. (2013-05). "Density functional theory description of electronic properties of wurtzite zinc oxide". Journal of Physics and Chemistry of Solids. 74 (5): 729–736. doi:10.1016/j.jpcs.2013.01.013. ISSN 0022-3697. {{cite journal}}: Check date values in: |date= (help)


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