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85037194376
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Note that the sum rule on the exchange-correlation hole is satisfied whatever the TDDFT kernel (Formula presented)
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Note that the sum rule on the exchange-correlation hole is satisfied whatever the TDDFT kernel (Formula presented)
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P. Nozières, Theory of Interacting Fermi Systems (Benjamin, New York, 1964).
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56
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85037195685
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By contrast, the “test charge-test charge” screened Coulomb interaction (Formula presented) yields the Coulomb potential felt by a test charge at (Formula presented) in response to another test charge at (Formula presented) Both the test charge-test charge and the electron-test charge screened Coulomb interactions are equal in the RPA, which does not include exchange-correlation effects
-
By contrast, the “test charge-test charge” screened Coulomb interaction (Formula presented) yields the Coulomb potential felt by a test charge at (Formula presented) in response to another test charge at (Formula presented) Both the test charge-test charge and the electron-test charge screened Coulomb interactions are equal in the RPA, which does not include exchange-correlation effects.
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57
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85037228303
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Note that proper infinitesimals must be included (as usual) in all poles when going from the imaginary to the real frequency axis. For example, the real frequency Green function (Formula presented) reads (Formula presented)where (Formula presented) is a positive infinitesimal and the sign (Formula presented) (resp. (Formula presented) holds for occupied (resp. empty) KS states. The analytical continuation of (Formula presented) from the positive imaginary frequency axis to the positive real frequency axis, and from the negative imaginary frequency axis to the negative real frequency axis, reads in the same way: (Formula presented)This is in fact the Fourier transform of the time-ordered (rather than causal) density-density response function
-
Note that proper infinitesimals must be included (as usual) in all poles when going from the imaginary to the real frequency axis. For example, the real frequency Green function (Formula presented) reads (Formula presented)where (Formula presented) is a positive infinitesimal and the sign (Formula presented) (resp. (Formula presented) holds for occupied (resp. empty) KS states. The analytical continuation of (Formula presented) from the positive imaginary frequency axis to the positive real frequency axis, and from the negative imaginary frequency axis to the negative real frequency axis, reads in the same way: (Formula presented)This is in fact the Fourier transform of the time-ordered (rather than causal) density-density response function.
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58
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85037180110
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T. Grabo, T. Kreibich, S. Kurth, and E.K.U. Gross, in The Strong Coulomb Correlations and Electronic Structure Calculations: Beyond the LDA, edited by V. Anisimov (Gordon and Breach, Amsterdam, 1999)
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T. Grabo, T. Kreibich, S. Kurth, and E.K.U. Gross, in The Strong Coulomb Correlations and Electronic Structure Calculations: Beyond the LDA, edited by V. Anisimov (Gordon and Breach, Amsterdam, 1999).
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59
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85037247208
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This restriction does not apply to the real frequency version of Eq. (32); it can be circumvented replacing (Formula presented) with (Formula presented) in Eq. (32)
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This restriction does not apply to the real frequency version of Eq. (32); it can be circumvented replacing (Formula presented) with (Formula presented) in Eq. (32).
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60
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85037228589
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C. Filippi, X. Gonze, and C.J. Umrigar, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996)
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C. Filippi, X. Gonze, and C.J. Umrigar, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).
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0000621478
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64
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85037237411
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What we call the Nozières functional in this paper is called the Luttinger-Ward functional in Ref. 22. Although the Nozières and Luttinger-Ward functional resemble each other, they yield different results away from self-consistency (see Sec. II C). The functional used in Ref. 22 is actually the one proposed by Nozières in Ref. 45
-
What we call the Nozières functional in this paper is called the Luttinger-Ward functional in Ref. 22. Although the Nozières and Luttinger-Ward functional resemble each other, they yield different results away from self-consistency (see Sec. II C). The functional used in Ref. 22 is actually the one proposed by Nozières in Ref. 45.
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65
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0000721099
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85037246225
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From simple first-order perturbation theory, we get (Formula presented) and (Formula presented) which should be quite accurate for the He atom. Hence the PGG polarizability of the He atom can be estimated as the average between the uncoupled (noninteracting) and RPA polarizabilities of Ref. 54, i.e., 1.37 a.u. Since (Formula presented) is half the PGG polarizability of the He atom, we finally get (Formula presented)
-
From simple first-order perturbation theory, we get (Formula presented) and (Formula presented) which should be quite accurate for the He atom. Hence the PGG polarizability of the He atom can be estimated as the average between the uncoupled (noninteracting) and RPA polarizabilities of Ref. 54, i.e., 1.37 a.u. Since (Formula presented) is half the PGG polarizability of the He atom, we finally get (Formula presented)
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70
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85037205400
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The exact polarizability of the hydrogen atom is (Formula presented) and scales like (Formula presented) for heavier hydrogenoid atoms. Hence (Formula presented)
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The exact polarizability of the hydrogen atom is (Formula presented) and scales like (Formula presented) for heavier hydrogenoid atoms. Hence (Formula presented)
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