Self-interaction correction to electronic structure ...

Faculty of Chemistry and Physics Institute of Theoretical Physics

Bachelor Thesis

Self-interaction correction to electronic structure calculations with density functional theory: Analytical second order derivatives in FLO SIC

bla Charlotte Vogelbusch

bla submitted: August 23, 2016 defended: September 2, 2016

September 16, 2016

Topic registered on:

11.05.2016 Tutor/First Proofreader:

Prof. Dr. Jens Kortus Second Proofreader:

Dr. Ronald Starke

Eidesstattliche Erklärung Ich versichere, dass ich diese Arbeit selbstständig verfasst und keine anderen Hilfsmittel als die angegebenen benutzt habe. Die Stellen der Arbeit, die anderen Werken dem Wortlaut oder dem Sinn nach entnommen sind, habe ich in jedem einzelnen Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht. Diese Versicherung bezieht sich auch auf die bildlichen Darstellungen.

16. September 2016

Declaration I hereby declare that I completed this work without any improper help from a third party and without using any aids other than those cited. All ideas derived directly or indirectly from other sources are identified as such. This declaration also refers to the representation of figures and visual material.

September 16, 2016

Contents

5

Contents 1. Introduction

7

2. Theoretical Background 2.1. Electronic Structure and Self-Interaction Error . . . . . . . . . . . . . . . . . 2.2. Perdew Zunger Self-Interaction Correction . . . . . . . . . . . . . . . . . . . . 2.3. Fermi-Löwdin Orbital Self-Interaction Correction . . . . . . . . . . . . . . . .

7 8 18 19

3. Derivatives 3.1. First Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Second Order Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Rearrangement of the Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 22 26 28

4. Conclusion

30

A. Appendix A.1. Improved Minimization using the Hessian . . . . . . . . . . . . . . . . . . . . A.2. Central Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33

References

35

: Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction

6

Contents

Abbreviations For the sake of a good reading flow, we introduce some abbreviations listed below. Abbreviation DFT SIC FLO SIC L(S)DA KS NRLMOL CG BFGS L-BFGS

Full size density functional theory self-interaction correction Fermi Löwdin orbital self-interaction correction local (spin) density approximation Kohn and Sham Naval Research Laboratory Molecular Orbital Library Conjugate Gradient Broyden-Fletcher–Goldfarb-Shanno Limited memory Broyden-Fletcher–Goldfarb-Shanno

Acknowledgment In the first place, I would like to thank Prof. Dr. Jens Kortus for introducing me to the topic of theoretical physics and the selection of my topic. Thank you for your sympathetic ear and your open mind. Most of all, I thank my second supervisor, Dr. Ronald Starke, for his eager eye, his precious time, his knowledge and accuracy and last but not least for the many informative discussions. Besides, I want to thank Lenz Fiedler, Dr. Torsten Hahn, B.Sc. René Wirnata and all other members of the Institute of Theoretical Physics for their host of advice, the interesting discussions and the tasty cake. A special thanks goes to my brother, B.A. Jonathan Vogelbusch for proofreading this document without understanding even one formula. Finally, I want to thank my family wholeheartedly for their mental and physical support, the regular divertissements and for being my earthwire. Thank you for bearing with me.


7

1. Introduction The calculation of materials properties is an interesting field since these properties are helpful for finding the right materials for various applications. Of course, there is the possibility of experimental analysis as well, but it is more expensive, time-consuming and in many cases not even possible. The prediction of properties, e.g. the stability under certain conditions, is one great advantage of theoretical investigations. All these ab-initio calculations are based on the Schrödinger equation. Since this equation is not separable and therefore not solvable for a few or more particles, approximations are required. Besides Hartee-Fock theory, one successful approximation is the density functional theory (DFT). It is the most widely used many-body approximation because of its short computing time and relatively good results. For example, heat capacities and surface electronic properties of metals and semi-conductors [24] can be well predicted with DFT. However, as every approximation, even DFT makes mistakes. One of them is the selfinteraction error. This deficiency of DFT leads e.g. to a wrong long-range behavior of the one-electron potential (not −1/𝑟), wrong total energies for metals and atoms [24], wrong level ordering [24] and problems with reduced band gaps [22]. For this reason, J. P. Perdew and A. Zunger introduced self-interaction correction (SIC) in 1981. They already achieved great successes, as e.g. correct −1/𝑟 long-range behavior. Moreover, DFT predicts the dissociation of molecules into spouriously charged fragments, which can be eliminated by Perdew Zunger SIC [27]. Unfortunately, the Perdew Zunger SIC scales with 𝑁 2 and is thus too computationally demanding for most applications. To circumvent this problem, M. R. Pederson et al. introduced a different version of SIC, the Fermi Löwdin orbital self-interaction correction (FLO SIC) [15, 16]. This method is treated in several publications, where FLO SIC is applied to various examples and the method is improved [4, 5, 11, 12, 13, 14, 19, 21]. The implementation of this method is done in the Naval Research Laboratory Molecular Orbital Library (NRLMOL) all electron computer code (see [6, 17, 18, 20, 23, 25, 26]). Unfortunately, the computational realization of the method is difficult, since the minimization of the SIC energy is demanding. An acceleration via preconditioning with second order derivatives was expected. For this purpose, this thesis gives the derivation of their expressions. For a fundamental understanding, we will first review many-body theory, especially DFT (Sec. 2.1). Next, we will outline the Perdew Zunger SIC (Sec. 2.2) and the FLO SIC method (Sec. 2.3). In Sec. 3 we will consider the first and second order derivatives of the SIC energy with respect to the Fermi orbital descriptors, that are required for the minimization of the SIC energy in the FLO SIC method. We start with a review of the derivation of the first derivative, given in [11, 12] (cp. Sec. 3.1). We will then derive the new expressions for the second order derivative in Secs. 3.2 and 3.3. For the convenience of the reader, the main content of this thesis will be summarized in Sec. 4. Finally, we will give an idea of the use of second order derivatives in numerical optimization algorithms in the appendix (Sec. A.1). For the sake of clarity, we then summarize the main expressions which are the result of this document in Sec. A.2.

2. Theoretical Background This section will outline the main aspects of many-body theory, DFT and SIC to gently lead the reader into the topic of this document: FLO SIC.


8

2. Theoretical Background

2.1. Electronic Structure and Self-Interaction Error The time evolution of every quantum system is determined by the Schrödinger equation which reads iℏ𝜕𝑡 |Ψ(𝑡)⟩ = 𝐻̂ |Ψ(𝑡)⟩ , (1) where i2 = −1, ℏ is the reduced Planck constant, 𝐻̂ and |Ψ⟩ denote the Hamiltonian and the quantum state. The coresponding eigenvalue problem for stationary systems reads 𝐻̂ |Ψ⟩ = 𝐸 |Ψ⟩ ,

(2)

which is the stationary Schrödinger equation. This equation constitutes an eigenvalue problem. Usually, we are interested in the ground state energy 𝐸0 , which is the lowest of these eigenvalues. The Hamiltonian for a system of 𝑍 nuclei and 𝑁 electrons is given by ̂ + 𝑉n-n ̂ + 𝑉e-e ̂ , 𝐻̂ = 𝑇ê + 𝑇n̂ + 𝑉e-n

(3)

̂ , where 𝑇ê and 𝑇n̂ are the kinetic energy operators of the electrons and the nuclei and 𝑉e-e ̂ ̂ 𝑉n-n and 𝑉e-n are the potential energy operators corresponding to the Coulomb interaction between the electrons, the nuclei and between both of them. Since it is too complicated to take the nuclei into account, they are set to fixed positions. This constitutes the fixed-nuclei approximation, that reduces the original problem to an exclusively electronic problem, whose Hamiltonian reads ̂ + 𝑉ext ̂n , 𝐻̂e = 𝑇ê + 𝑉e-e (4) ̂ n is the potential energy of the electrons in the external potential generated by the where 𝑉ext fixed nuclei. Due to their importance, the operators contributing to the electronic structure Hamiltonian 𝐻̂𝑒 will be spelled out explicitly in the following. As a matter of principle, the one-particle kinetic energy operator reads 𝑇̂ = −

ℏ2 Δ. 2𝑚

(5)

For 𝑁 electrons, this generalizes to 𝑁

ℏ2 Δ , 2𝑚e 𝑖 𝑖=1

𝑇ê = − ∑

(6)

where Δ𝑖 is the Laplace operator that acts on the coordinates of the 𝑖-th electron and 𝑚e is the electron mass. We will now review the Coulomb interaction. Classically, the interaction between two charged particles is described by Coulomb’s law, which reads (see [1, Eq. 5.1]) 𝑭12 =

1 𝒙 − 𝒙2 . 𝑞1 𝑞2 1 4𝜋𝜀0 |𝒙1 − 𝒙2 |3

(7)

Here, 𝑭12 is the force of the particle with charge 𝑞2 on the position 𝒙2 exerted on the particle with charge 𝑞1 on the position 𝒙1 . Finally, 𝜀0 is the vacuum permittivity. We can also interpret 𝑭12 as generated by the electric field 𝑬2 of the particle with charge 𝑞2 , i.e. 𝑭12 = 𝑞1 𝑬2 ,

(8)


2.1. Electronic Structure and Self-Interaction Error

9

with the electrical field 𝑬2 given by 𝑬2 (𝒙1 ) =

𝒙 − 𝒙2 1 𝑞2 1 . 4𝜋𝜀0 |𝒙1 − 𝒙2 |3

(9)

Analogously, the electric field generated by a continuous charge density 𝜌(𝒙) reads (cf. e.g. [1, Eq. 5.19]) 𝒙 − 𝒙′ 1 ∫ d3 𝒙′ 𝜌(𝒙′ ) . (10) 𝑬(𝒙) = 4𝜋𝜀0 |𝒙 − 𝒙′ |3 This field is longitudinal, i.e. ∇×𝑬 =0,

(11)

and can hence be expressed as a gradient, 𝑬(𝒙) = −∇𝜑(𝒙) .

(12)

It can be shown easily, that the Coulomb potential (cf. e.g. [1, Eq. 6.3]) 𝜑(𝒙) =

𝜌(𝒙′ ) 1 ∫ d3 𝒙′ 4𝜋𝜀0 |𝒙 − 𝒙′ |

(13)

reproduces Eq. (10) via Eq. (12), where one has to use the relation ∇

𝒙 − 𝒙′ 1 = − . |𝒙 − 𝒙′ | |𝒙 − 𝒙′ |3

(14)

For the sake of simplicity, we define the Coulomb integral kernel 𝑣(𝒙 − 𝒙′ ) as 𝑣 (𝒙 − 𝒙′ ) =

1 1 . 4𝜋𝜀0 |𝒙 − 𝒙′ |

(15)

With that, Eq. (13) reads 𝜑(𝒙) = ∫ d3 𝒙′ 𝑣 (𝒙 − 𝒙′ )𝜌(𝒙′ ) .

(16)

On the other hand, the electric field energy 𝑉 can be written as the integral over the squared field amplitude as 𝜀 𝑉 = 0 ∫ d3 𝒙 𝑬(𝒙) ⋅ 𝑬(𝒙) . (17) 2 Using Eq. (12), this can be rewritten as 𝑉 =

𝜀0 ∫ d3 𝒙 ∇𝜑(𝒙) ⋅ ∇𝜑(𝒙) . 2

(18)

By partial integration, we get 𝑉 =−

𝜀0 ∫ d3 𝒙 𝜑(𝒙)Δ𝜑(𝒙) . 2

(19)

With help of Poisson’s equation, −Δ𝜑(𝒙) =

𝜌(𝒙) , 𝜀0


(20)

10


we finally obtain

1 ∫ d3 𝒙 𝜑(𝒙)𝜌(𝒙) . (21) 2 With this relation and Eq. (13), the electric field energy 𝑉 of the charge density 𝜌 reads (cp. e.g. [1, Eq. 6.29]) 𝑉 =

𝑉 =

1 ∫ d3 𝒙∫ d3 𝒙′ 𝜌(𝒙)𝑣 (𝒙 − 𝒙′ )𝜌(𝒙′ ) . 2

(22)

This is the Coulomb self-energy of a charge density in its own electric field. We now consider a charge density 𝜌, that is the sum of two charge densities 𝜌1 and 𝜌2 , i.e. 𝜌(𝒙) = 𝜌1 (𝒙) + 𝜌2 (𝒙) .

(23)

For the electric field energy we now obtain 1 ∫ d3 𝒙∫ d3 𝒙′ 𝜌(𝒙)𝑣 (𝒙 − 𝒙′ )𝜌(𝒙′ ) 2 1 = ∫ d3 𝒙∫ d3 𝒙′ 𝜌1 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌1 (𝒙′ ) + 2 1 + ∫ d3 𝒙∫ d3 𝒙′ 𝜌2 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌1 (𝒙′ ) + 2 1 = ∫ d3 𝒙∫ d3 𝒙′ 𝜌1 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌1 (𝒙′ ) + 2

𝑉 =

1 ∫ d3 𝒙∫ d3 𝒙′ 𝜌1 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌2 (𝒙′ ) 2 1 ∫ d3 𝒙∫ d3 𝒙′ 𝜌2 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌2 (𝒙′ ) 2 1 ∫ d3 𝒙∫ d3 𝒙′ 𝜌2 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌2 (𝒙′ ) 2

+ ∫ d3 𝒙∫ d3 𝒙′ 𝜌1 (𝒙)𝑣 (𝒙 − 𝒙′ )𝜌2 (𝒙′ ) (24)

= 𝑉11 + 𝑉22 + 𝑉12 .

The first two terms are the self-energies of the charge densities 𝜌1 and 𝜌2 . The third term is the potential energy of the charge density 𝜌1 in the external field created by the charge density 𝜌2 and vice versa. Explicitly, this potential energy reads (cp. [1, Eq. 6.28]) 𝑉12 = ∫ d3 𝒙∫ d3 𝒙′ 𝜌1 (𝒙) 𝑣 (𝒙 − 𝒙′ )𝜌2 (𝒙′ ) .

(25)

Concretely, we now consider 𝑁 point charges with charge 𝑞 and positions 𝒙𝑖 . The corresponding charge density 𝜌 reads (cf. [28, Eq. 2.9]) 𝑁

𝜌(𝒙) = 𝑞 ∑ 𝛿 3 (𝒙 − 𝒙𝑖 ) ,

(26)

𝑖=1

where 𝛿 3 (𝒙 − 𝒙𝑖 ) is the Dirac delta distribution. The resulting electric field energy is then given by 𝑉 =

𝑁 1 𝑁 1 1 ∑ 𝑣 (𝒙𝑖 − 𝒙𝑗 ) = ∑ . 2 𝑖,𝑗=1 8𝜋𝜀0 𝑖,𝑗=1 ∣𝒙𝑖 − 𝒙𝑗 ∣

(27)

We note, that even terms were 𝑖 = 𝑗 contribute to the energy. Evidently, these becomes singular, which means that the classical Coulomb self-energy of a point-particle is infinite. In order to obtain a finite result, the electric field energy is redefined as 𝑉 ∶=

𝑁 1 1 ∑ , 8𝜋𝜀0 𝑖,𝑗=1 ∣𝒙𝑖 − 𝒙𝑗 ∣

(28)

𝑖≠𝑗



11

i.e. terms whit 𝑖 = 𝑗 are excluded. In quantum mechanics, classical observables are replaced by their corresponding operators by quantization. Formally, the Coulomb interaction operator 𝑉 ̂ would then read 1 𝜌(𝒙) ̂ 𝜌(𝒙 ̂ ′) ? 𝑉̂ = ∫ d3 𝒙∫ d3 𝒙′ (29) 8𝜋𝜀0 |𝒙 − 𝒙′ | in terms of the charge density operator 𝜌.̂ (cf. Eq. (26) with 𝑞 = −𝑒)

For 𝑁 electrons, this operator is given by 𝑁

𝜌(𝒙) ̂ = (−𝑒) ∑ 𝛿 3 (𝒙 − 𝒙̂ 𝑖 ) .

(30)

𝑖=1

This would result in the Coulomb interaction operator (see [28, Eq. 2-19]) ?

𝑉̂ =

(−𝑒)2 𝑁 1 (−𝑒)2 𝑁 ∑ = ∑ 𝑣 (𝒙̂ 𝑖 − 𝒙̂ 𝑗 ) , 8𝜋𝜀0 𝑖,𝑗=1 ∣𝒙̂ 𝑖 − 𝒙̂ 𝑗 ∣ 2 𝑖,𝑗=1

(31)

which includes self-interaction terms. However, exactly as in the classical case, the selfinteraction terms of the point particles have to be discarded such that the true Coulomb interaction operator reads: 𝑉̂ =

1 (−𝑒)2 𝑁 (−𝑒)2 𝑁 ∑ = ∑ 𝑣 (𝒙̂ 𝑖 − 𝒙̂ 𝑗 ) . 8𝜋𝜀0 𝑖,𝑗=1 ∣𝒙̂ 𝑖 − 𝒙̂ 𝑗 ∣ 2 𝑖,𝑗=1 𝑖≠𝑗

(32)

𝑖≠𝑗

In the quantum field theoretical context, this exclusion of self-interaction terms is put into practice by the so called normal ordering [7, Eq. 3.110], denoted by colons: ̂ = 𝑉 ̂ = 𝑉e-e

∶ 𝜌(𝒙) ̂ 𝜌(𝒙 ̂ ′) ∶ 1 ∫ d3 𝒙∫ d3 𝒙′ . 8𝜋𝜀0 |𝒙 − 𝒙′ |

(33)

After this review of the Coulomb interaction, we will now consider the state space of the many-body Hamiltonian. The 𝑁 -electron Hilbert space is constructed from the 𝑁 oneelectron Hilbert spaces ℋ as the 𝑁 -fold tensor product ℋ𝑁 = ℋ ⊗ ℋ ⊗ ... ⊗ ℋ . ⏟⏟⏟⏟⏟⏟⏟

(34)

𝑁 factors

An orthonormal set of basis functions |𝜓𝑖 ⟩ in the one-particle Hilbert space ℋ induces a basis of the 𝑁 -particle Hilbert space, which can alternatively be written as |Ψ𝑖1 …𝑖𝑁 ⟩ = |𝜓𝑖1 ⟩ ⊗ |𝜓𝑖2 ⟩ ⊗ ... ⊗ |𝜓𝑖𝑁 ⟩ , ≡ |𝜓𝑖1 ⊗ 𝜓𝑖2 ⊗ ... ⊗ 𝜓𝑖𝑁 ⟩ , ≡ |𝜓𝑖1 ⟩ |𝜓𝑖2 ⟩ ... |𝜓𝑖𝑁 ⟩ , ≡ |𝜓𝑖1 𝜓𝑖2 ...𝜓𝑖𝑁 ⟩ .

(35)

A so called one-particle operator in the 𝑁 -particle state space is of the form 𝐻̂ = 𝐻̂1 ⊗ ⏟ 1⏟ ⊗⏟ ... ⏟ ⊗⏟ 1 +1 ⊗ 𝐻̂1 ⊗ ⏟ 1⏟ ⊗⏟ ... ⏟ ⊗⏟ 1+⋯ + ⏟ 1⏟ ⊗⏟ ... ⏟ ⊗⏟ 1 ⊗𝐻̂1 , 𝑁−1

𝑁−2

𝑁−1


(36)

12


where 1 is the identity operator in the one-particle state space. I.e. a one-particle operator acts on each one-particle orbital separately and the terms have to be summed up. For ̂ acts on a 𝑁 -particle wave function as example, a one-particle multiplication operator 𝑉ext ̂ Ψ) (𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) = 𝑣ext (𝒙1 ) Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) + 𝑣ext (𝒙2 ) Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) (𝑉ext + ... + 𝑣ext (𝒙𝑁 ) Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) .

(37)

We now consider the 𝑁 -particle stationary Schrödinger equation 𝐻̂ |Ψ⟩ = 𝐸 |Ψ⟩ ,

(38)

where 𝐻̂ is a one-particle operator as in Eq. (36). Furthermore, we assume that we can solve the one-particle eigenvalue problems 𝐻̂1 |𝜓𝑖 ⟩ = 𝜖𝑖 |𝜓𝑖 ⟩ .

(39)

A solution to the many-body Schrödinger equation is then given by Eq. (35). Correspondingly, the eigenenergies of 𝐻̂ are given as the sum of the one-electron eigenenergies 𝜖𝑖 , which reads 𝑁

𝐸 = ∑ 𝜖𝑖 .

(40)

𝑖=1

Put differently, if the Hamiltonian of the many-body system is non-interacting (i.e. given by a one-particle operator), the many-body Schrödinger equation separates into one-particle problems. The Coulomb potential operator is not a one-particle operator. Instead, it acts on a 𝑁 electron wave function as follows 𝑁

1 (𝑉 ̂ Ψ) (𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) = ∑ 𝑣(𝒙̂ 𝑖 − 𝒙̂ 𝑗 )Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) 2 𝑖,𝑗=1 𝑖≠𝑗

= 𝑣(𝒙̂ 1 − 𝒙̂ 2 )Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) + 𝑣(𝒙̂ 1 − 𝒙̂ 3 )Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) + ... + 𝑣(𝒙̂ 𝑁−1 − 𝒙̂ 𝑁 )Ψ(𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) .

(41)

We now evaluate the expectation value of the Coulomb interaction operator in various 𝑁 particle states. The simplest example is a two-particle product state given by ⟨𝒙1 𝒙2 ∣Φ2 ⟩ = Φ2 (𝒙1 , 𝒙2 ) = 𝜓1 (𝒙1 ) 𝜓2 (𝒙2 ) .

(42)

For this wave function, the charge density 𝜌 is the sum of the densities of the one-particle orbitals 𝜌1 and 𝜌2 . Explicitly, it reads 𝜌(𝒙) = 𝜌1 (𝒙) + 𝜌2 (𝒙) = (−𝑒)𝜓1∗ (𝒙) 𝜓1 (𝒙) + (−𝑒)𝜓2∗ (𝒙) 𝜓2 (𝒙) .

(43)

The expectation value of the Coulomb operator in the product state is then given by (−𝑒)2 1 ⟨Φ2 |𝑉 ̂ |Φ2 ⟩ = 𝜓 (𝒙 ) 𝜓 (𝒙 ) ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 4𝜋𝜀0 |𝒙1 − 𝒙2 | 1 1 2 2 = (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓1 (𝒙1 ) 𝜓2 (𝒙2 ) = (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓1 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓2∗ (𝒙2 ) 𝜓2 (𝒙2 ) = ∫ d3 𝒙1 ∫ d3 𝒙2 𝜌1 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌2 (𝒙2 ) .

(44)



13

By contrast, the classical Coulomb self-energy (cp. Eq. (22)) of the density 𝜌 of this product state |Φ2 ⟩ is given by (cp. Eq. (24)) 1 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜌1 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌1 (𝒙2 ) 2 1 + ∫ d3 𝒙1 ∫ d3 𝒙2 𝜌2 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌2 (𝒙2 ) 2

𝑉 (𝜌) =

+ ∫ d3 𝒙1 ∫ d3 𝒙2 𝜌1 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌2 (𝒙2 ) .

(45)

We now compare the expactation value of the Coulomb operator ⟨Φ2 |𝑉 ̂ |Φ2 ⟩ (see Eq. (44)) and the classical Coulomb potential 𝑉 (𝜌) (see Eq. (45)) to get ⟨Φ2 |𝑉 ̂ |Φ2 ⟩ = 𝑉 (𝜌) − −

1 ∫ d3 𝒙1∫ d3 𝒙2 𝜌1 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌1 (𝒙2 ) 2

1 ∫ d3 𝒙1∫ d3 𝒙2 𝜌2 (𝒙1 ) 𝑣 (𝒙1 − 𝒙2 )𝜌2 (𝒙2 ) . 2

(46)

Apparently, the classical expression includes the self-energies for 𝜌1 and 𝜌2 . These selfinteraction terms do not appear in the quantum mechanical description and therefore have to be discarded. We will now extend our discussion to antisymmetric wave functions, because electrons are fermions and hence their wave functions have to be antisymmetric. For this reason, the basis of the 𝑁 -electron Hilbert space is not given by Eq. (35), but by the corresponding Slater determinants |Ψ⟩, which are the antisymmetrizised and normalized versions of the product states given in Eq. (35). Explicitly, they read 𝜓 (𝒙 ) 𝜓2 (𝒙1 ) ∣ 1 1 1 ∣ 𝜓1 (𝒙2 ) 𝜓2 (𝒙2 ) Ψ𝑁 (𝒙1 , 𝒙2 , ..., 𝒙𝑁 ) = √ ⋮ ⋮ 𝑁! ∣ 𝜓 (𝒙 ) 𝜓 (𝒙 ∣ 1 𝑁 2 𝑁)

⋯ 𝜓𝑁 (𝒙1 ) ∣ ⋯ 𝜓𝑁 (𝒙2 ) ∣ ⋱ ⋮ ∣ ⋯ 𝜓𝑁 (𝒙𝑁 )∣

(47)

for the 𝑁 -electron wave function constructed from the one-electron orbitals |𝜓1 ⟩ , … , |𝜓𝑁 ⟩. For 𝑁 = 2 the wave function |Ψ2 ⟩ reads in the position state basis 1 ⟨𝒙1 𝒙2 ∣Ψ2 ⟩ = Ψ2 (𝒙1 , 𝒙2 ) = √ (𝜓1 (𝒙1 )𝜓2 (𝒙2 ) − 𝜓2 (𝒙1 )𝜓1 (𝒙2 )) . 2

(48)

The expectation value of the Coulomb operator 𝑉 ̂ (cp. Eq. (32)) in this two-electron state is given by ⟨Ψ2 |𝑉 ̂ |Ψ2 ⟩ = (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓1 (𝒙1 ) 𝜓2 (𝒙2 ) − (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓1 (𝒙2 ) 𝜓2 (𝒙1 ) .

(49)

The first term is called Hartree term, the second one exchange term. As a matter of principle, the charge density resulting from a Slater determinant is also given as the sum of the corresponding one-orbital charge densities, i.e. 𝜌(𝒙) = ∑ 𝜌𝑖 (𝒙) = (−𝑒) ∑ 𝜓𝑖∗ (𝒙) 𝜓𝑖 (𝒙) . 𝑖

𝑖


(50)

14


Consequently, the Coulomb self-energy or Hartree term of the two-electron Slater determinant |Ψ2 ⟩ is the same as for the product state |Φ2 ⟩ given in Eq. (45). The comparison of the respective expectation values yields ⟨Ψ2 |𝑉 ̂ |Ψ2 ⟩ = ∫ d3 𝒙1 ∫ d3 𝒙2 𝜌1 (𝒙) 𝑣 (𝒙 − 𝒙′ ) 𝜌2 (𝒙) − (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓1 (𝒙2 ) 𝜓2 (𝒙1 ) = ⟨Φ2 |𝑉 ̂ |Φ2 ⟩ − (−𝑒)2 ∫ d3 𝒙1 ∫ d3 𝒙2 𝜓1∗ (𝒙1 ) 𝜓2∗ (𝒙2 ) 𝑣 (𝒙1 − 𝒙2 ) 𝜓1 (𝒙2 ) 𝜓2 (𝒙1 ) ,

(51)

i.e., the Hartree term equals the classical Coulomb interaction energy corresponding to the charge densities 𝜌1 and 𝜌2 . By contrast, the exchange term does not have any classical analogon. For later purposes, we introduce the Coulomb interaction matrix elements 𝑉𝑖𝑗𝑘𝑙 = (−𝑒)2 ∫ d3 𝒙∫ d3 𝒙′ 𝜓𝑖∗ (𝒙) 𝜓𝑗∗ (𝒙′ ) 𝑣 (𝒙 − 𝒙′ ) 𝜓𝑘 (𝒙)𝜓𝑙 (𝒙′ ) .

(52)

Apparently, these terms yield the same results, if 𝑖 and 𝑗 and 𝑘 and 𝑙 are interchanged at the same time. Explicitly, this symmetry can be written as 𝑉𝑖𝑗𝑘𝑙 = 𝑉𝑗𝑖𝑙𝑘 .

(53)

⟨Ψ2 ∣𝑉 ̂ ∣Ψ2 ⟩ = 𝑉1212 − 𝑉1221 .

(54)

With 𝑉𝑖𝑗𝑘𝑙 Eq. (49) reads

Generally, the expectation value of the Coulomb interaction operator in a 𝑁 -electron Slater determinant |Ψ𝑁 ⟩ (Eq. (47)) can now be written as 1 𝑁 ⟨Ψ𝑁 ∣𝑉 ̂ ∣Ψ𝑁 ⟩ = ∑ (𝑉 − 𝑉𝑖𝑗𝑗𝑖 ) , 2 𝑖,𝑗=1 𝑖𝑗𝑖𝑗

(55)

where the factor 1/2 appears because of the symmetry of 𝑉𝑖𝑗𝑘𝑙 . Here, the first term corresponds to the Hartree contribution while the second term reproduces the exchange energy. We note in particular, that it is not necessary to exclude contributions where 𝑖 = 𝑗 as in Eq. (33), since in this case the Hartree and the exchange term cancel. Explicitly, these terms read 𝑉𝑖𝑖𝑖𝑖 = ∫ d3 𝒙∫ d3 𝒙′ 𝜌𝑖 (𝒙) 𝑣 (𝒙 − 𝒙′ ) 𝜌𝑖 (𝒙′ ) . (56) With this, the first term in Eq. (55) can be written as 1 𝑁 ∑ 𝑉 = 𝑉 (𝜌) . 2 𝑖,𝑗=1 𝑖𝑗𝑖𝑗

(57)

In other words, this expression equals the classical Coulomb interaction energy of the charge density 𝜌 that corresponds to the Slater determinant |Ψ𝑁 ⟩. As a matter of principle, this contribution therefore also includes the self-interaction energies of each electron with itself. In the end, however, these are canceled out by the exchange term. : Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction


15

̂ Since the electronic structure Hamiltonian contains the Coulomb interaction operator 𝑉e-e that is no one-electron operator, the corresponding Schrödinger equation is not separable. The problem is hence too complex to be solved directly and one therefore needs to find independent-particle approximations, i.e. approximations that involve only one-electron operators. One of these is DFT, where the energy can be regarded as a functional of the electron density in the ground state. The one-electron reduced density matrix reads 𝑛(𝒙, 𝒙′ ) = 𝑁 ∫ d3 𝒙2 ...∫ d3 𝒙𝑁 Ψ∗ (𝒙, 𝒙2 , ... , 𝒙𝑁 ) Ψ(𝒙′ , 𝒙2 , ... , 𝒙𝑁 ) .

(58)

The electron density in the ground state is thus defined as (cp. [2, Eq. 7.7]) 𝑛(𝒙) = ⟨Ψ|𝑛(𝒙)|Ψ⟩ ̂ = 𝑁 ∫ d3 𝒙2 ...∫ d3 𝒙𝑁 Ψ∗ (𝒙, 𝒙2 , ... , 𝒙𝑁 ) Ψ(𝒙, 𝒙2 , ... , 𝒙𝑁 ) = 𝑛(𝒙, 𝒙) . (59) This electron density 𝑛 is related to the charge density 𝜌 of the electron liquid by the simple equation 𝜌(𝒙) = (−𝑒)𝑛(𝒙) . (60) Analogously, this relation is valid for the operators 𝑛̂ and 𝜌.̂ In the following, we will understand 𝑛 as the ground state density, where |Ψ⟩ is the ground state of the considered Hamiltonian. For a fundamental understanding of DFT, the concept of classes of Hamiltonians is required. We define a class of Hamiltonians as a set of Hamiltonians that only differ in their ̂ . The simplest class of Hamiltonians contains only this external poexternal potential 𝑉ext tential, i.e. ̂ . 𝐻̂ = 𝑇 ̂ + 𝑉ext (61) Another class of Hamiltonians is for example given by different electronic structure Hamiltonians 𝐻̂e ̂ + 𝑉ext ̂ , 𝐻̂e = 𝑇ê + 𝑉e-e (62) with equal kinetic energy and Coulomb interaction potential operators. With that concept, the idea of DFT can be described as follows: DFT is based on the fundamental theorems, the Hohenberg-Kohn theorems [8, p. 122ff.]. Hohenberg and Kohn proved, that the density of the ground state of an electronic structure Hamiltonian determines the external potential uniquely within its class. Furthermore, the ̂ a universal energy second Hohenberg-Kohn theorem says that for every external potential 𝑉ext functional 𝐸[𝑛] depending on the density 𝑛 can be defined, whose global minimum is the exact ground state energy 𝐸0 for any particular external potential. The density that minimizes the functional is the exact ground state density 𝑛. From these theorems it follows that all properties of the system are completely determined by the ground state density 𝑛 and that the energy functional 𝐸[𝑛] determines the ground state energy and density, whereas excited states require a separate theory. Nonetheless, the Hohenberg-Kohn theorems do not per se provide a solution to the problem of the Schrödinger equation for a 𝑁 -electron system. For this reason, Kohn and Sham (KS) introduced an auxiliary system of non-interacting particles, where the Hamiltonian consists of one-particle operators such that the Schrödinger equation can be separated [8, p. 136]. They related both Hamiltonians with the Hohenberg-Kohn theorems, so the auxiliary system yields the same results as the correct system. The following Scheme (Fig. 1) is based on [8, Figure 7.1] and illustrates this idea of Kohn and Sham.


16


̂ (𝒙) 𝑉ext

Hohenberg-Kohn

Eq. (4) 𝐻̂e

Eq. (59) Eq. (2)

Hohenberg-Kohn

𝑛(𝒙)

̂ (𝒙) 𝑉KS

Eq. (65)

𝜓𝑖 (𝒙)

Ψ(𝒙1 , ... , 𝒙𝑁 )

Eq. (63) 𝑁 ⋅ Eq. (2)

𝐻̂KS

Fig. 1: Scheme to outline the Kohn-Sham ansatz

We now regard two different classes of Hamiltonians. On the one hand, we consider the standard electronic structure Hamiltonian 𝐻̂e (given in Eq. (4)), that contains the external ̂ (left hand side). On the other hand, we consider a free Hamiltonian 𝐻̂KS , that potential 𝑉ext ̂ (right hand side). Explicitly, it reads (see [8, Eq. 7.1]) contains only an external potential 𝑉KS ̂ . 𝐻̂KS = 𝑇ê + 𝑉KS

(63)

Since these Hamiltonians differ in more than the external potential, they are of two different classes. For this reason, the same ground state density 𝑛 determines the different external ̂ and 𝑉KS ̂ at once, each in its class of Hamiltonians. potentials 𝑉ext ̂ and therefore the electronic Hamiltonian 𝐻̂e . The solution On the one hand, 𝑛 gives 𝑉ext of the related stationary Schrödinger equation yields the eigenfunctions and thus the ground state wave function |Ψ⟩. According to Eq. (59), the ground state density can be recalculated from |Ψ⟩. ̂ of the auxiliary system and therefore its Hamiltonian On the other hand, 𝑛 determines 𝑉KS 𝐻̂KS . In this non-interacting auxiliary quantum system, one simply has to solve 𝑁 oneelectron stationary Schrödinger equations like 𝐻̂KS |𝜓𝑖 ⟩ = 𝜖𝑖 |𝜓𝑖 ⟩ .

(64)

Thus, one finds the Kohn-Sham eigenvalues 𝜖𝑖 and the Kohn-Sham orbitals |𝜓𝑖 ⟩. Since the 𝑁 -electron wave function is the Slater determinant (see Eq. (47)) of the Kohn-Sham orbitals |𝜓𝑖 ⟩, the ground state density of this auxiliary system, that equals per constructionem the ground state density of the original system, reads in terms of the Kohn-Sham orbitals 𝑁

𝑁

𝑛(𝒙) = ∑ 𝑛𝑖 (𝒙) = ∑ 𝜓𝑖∗ (𝒙)𝜓𝑖 (𝒙) . 𝑖=1

(65)

𝑖=1

In this way we straightforwardly get to the ground state density of the original problem by solving one-electron Schrödinger equations. However, one now has to solve a self-consistent problem, i.e. one calculates a density 𝑛 as a ̂ , that depends itself on this density 𝑛 (cp. Eq. (69)). In function of an external potential 𝑉ext practice, one starts with an initial guess for a set of Kohn-Sham orbitals and calculates their ground state density. Therefrom, one then gets the external potential and the Hamiltonian. By solving the stationary Schrödinger equations, one gets new Kohn-Sham orbitals. This procedure is iterated until the change in the Kohn-Sham orbitals is sufficiently small.



17

We remember that the original problem was to find the ground state energy 𝐸0 of our system. This can be directly calculated from the ground state density. For this purpose, an expression of the energy functional is required. Explicitly, it reads [8, Eq. 7.5] 𝐸0 [𝑛] = 𝑇s [𝑛] + 𝐸ext [𝑛] + 𝐸H [𝑛] + 𝐸n-n [𝑛] + 𝐸xc [𝑛] ,

(66)

where 𝑇s is the independent-particle kinetic energy (see [8, Eq. 7.3]), 𝐸ext is the external energy, 𝐸H is the Hartree energy, 𝐸n-n is the interaction energy of the nuclei and 𝐸xc is the exchange-correlation energy. The external energy is given by 𝐸ext [𝑛] = ∫ d3 𝒙 𝑣ext (𝒙) 𝑛(𝒙) .

(67)

The Hartree energy in terms of the density reads (cf. [8, Eq. 7.4]) 𝐸H [𝑛] =

𝑛(𝒙)𝑛(𝒙′ ) (−𝑒)2 ∫ d 3 𝒙 ∫ d 3 𝒙′ . 8𝜋𝜀0 |𝒙 − 𝒙′ |

(68)

It describes the interaction energy of the electron density with itself and is equal to the Coulomb self-energy (cp. Eqs. (22) and (60)). As in the classical expression, there is no exclusion of the self-interaction. Analogously to Eq. (49), the exchange-correlation energy ought to cancel out the self-interaction exactly. Its correct expression, however, is unknown. ̂ has not We will now return to the Kohn-Sham Hamiltonian (see Eq. (63)), where 𝑉KS been defined yet, as it depends on the energy functional that had to be introduced in the ̂ is a multiplication operator and acts on a first place. Per definitionem, the operator 𝑉KS 𝑁 -electron wave function as defined in Eq. (37). The corresponding external potential 𝑣KS is given by [8, Eq. 7.13] 𝛿 𝐸H 𝛿 𝐸xc 𝛿 𝐸ext + + 𝛿 𝑛(𝒙) 𝛿 𝑛(𝒙) 𝛿 𝑛(𝒙) = 𝑣ext (𝒙) + 𝑣H (𝒙) + 𝑣xc (𝒙) ,

𝑣KS (𝒙) =

(69)

with the Hartree potential 𝑣H [𝑛](𝒙) =

(−𝑒)2 𝑛(𝒙′ ) ∫ d 3 𝒙′ . 8𝜋𝜀0 |𝒙 − 𝒙′ |

(70)

Remarkably, these operators and thereby the whole Hamiltonian 𝐻̂KS depends on the density that in turn has to be calculated from its own eigenfunctions. Hence, the eigenvalue problem corresponding to 𝐻̂KS is no longer linear. In principle, DFT is an exact many-body theory (apart from the fixed-nuclei approximation). In practice, however, the expression for the exchange correlation energy 𝐸xc and hence ̂ has to be approximated. for the corresponding operator 𝑉xc There are several approximations for 𝐸xc , starting with the local (spin) density approximation (L(S)DA) and leading to various hybrid orbitals that combine DFT with other manybody theories like Hartree-Fock. Until today, new functionals are developed over and over again. But still, if the exact exchange-correlation functional is not known, it can not cancel out the self-interaction of the Hartree term completely. We will come back to this issue in the next section.


18


But first, we close this section with a technical remark: In this document, the spin index 𝜎 ∈ {↑, ↓} has been suppressed. If desired, it can be regained by the following replacement rules: 𝜓(𝒙) ↦ 𝜓𝜎 (𝒙) = (

𝜓↑ (𝒙) ) , 𝜓↓ (𝒙)

† 𝜓∗ (𝒙) ↦ 𝜓𝜎 (𝒙) = (𝜓↑∗ (𝒙), 𝜓↓∗ (𝒙)) ,

∫ d3 𝒙 ↦ ∑ ∫ d3 𝒙 , 𝜎=↑,↓

̂ ′, 𝐴 ̂ ↦ 𝐴𝜎𝜎 ̂ ↦ ∑ 𝐴̂ ′ 𝜓 ′ . 𝐴𝜓 𝜎𝜎 𝜎

(71)

𝜎′ =↑,↓

Similarly, the one-particle Hilbert space is now given by ℋ = 𝐿2 ↦ ℋ = ℂ 2 ⊗ 𝐿 2 .

(72)

For example, the spin resolved density, 𝑛𝜎 (𝒙) (see [8, Eq. 7.2]), replaces 𝑛(𝒙) (cp. Eq. (59)) as 𝑁

𝑁

∗ 𝑛(𝒙) = ∑ 𝜓𝑖∗ (𝒙)𝜓𝑖 (𝒙) ↦ 𝑛(𝒙) = ∑ 𝑛𝜎 (𝒙) = ∑ ∑ 𝜓𝑖𝜎 (𝒙)𝜓𝑖𝜎 (𝒙) . 𝑖=1

𝜎′ =↑,↓

(73)

𝜎 𝑖=1

Furthermore, the scalar product ⟨𝜑|𝜓⟩ has to be replaced as ([28, Eq. B.127]) ∫ d3 𝒙 𝜑∗ (𝒙)𝜓(𝒙) ↦ ∫ d3 𝒙 𝜑† (𝒙)𝜓(𝒙) = ∫ d3 𝒙 (𝜑∗↑ (𝒙)𝜓↑ (𝒙) + 𝜑∗↓ (𝒙)𝜓↓ (𝒙)) .

(74)

In this way, in case of interest all spin resolved expressions can be gained back from their scalar counterparts given in this document.

2.2. Perdew Zunger Self-Interaction Correction This section will treat the self-interaction error and its correction as proposed by J. P. Perdew and A. Zunger in 1981 [24]. Self-interaction originally occurred when quantizing the classical Coulomb energy (cp. Sec. 2.1). The expectation value of the Coulomb operator in a 𝑁 -electron wave function excludes all terms, that depend on the same orbital, i.e. the self-interaction terms. By contrast, this is not the case for the classical Coulomb energy. In DFT, this problem returns because the energy functional 𝐸[𝑛] includes the Hartree energy (see Eq. (68)). Theoretically, DFT avoids the self-interaction error, whilst the exact exchange-correlation energy cancels the self-interaction of the Hartree energy. However, as no exact description of the exchange-correlation functional is known, it has to be approximated. Unfortunately, the current approximations do not cancel the self-interaction. For this reason, DFT as implemented today entails a self-interaction error. J. P. Perdew and A. Zunger observed that for a correct (i.e. self-interaction free) description, the exchange-correlation energy 𝐸xc of a single, fully occupied orbital 𝑛𝑖 must exactly cancel its Hartree energy 𝐸H . Explicitly, this means (see [24, Eq. 29]) 𝐸H [𝑛𝑖 ] + 𝐸xc [𝑛𝑖 ] = 0 .

(75)

Here, 𝑛𝑖 denotes the electronic density of the respective orbital (cp. Eq. (65)) 𝑛𝑖 (𝒙) = |𝜓𝑖 (𝒙)|2 .

(76)


2.3. Fermi-Löwdin Orbital Self-Interaction Correction

19

For the correct exchange-correlation energy, Eq. (75) is satisfied. Unfortunately, its expression is not known. Approximations to 𝐸xc achieve only partial cancellation. Thus a residue of the self-interaction error remains in all recent DFT calculations. Several approximations to the exchange-correlation functional try to get rid of the self-interaction error. These “[e]mpirical methods” [24, p. 23] become unnecessary, when the SIC is applied to the system. app Perdew and Zunger suggested a SIC to an approximation 𝐸xc by subtraction of the sum of SIC the SIC energies 𝐸 [𝑛𝑖 ] of each orbital 𝑖. The self-interaction corrected exchange correlation cor energy 𝐸xc is thus given by [24, Eq. 32; notation adapted] 𝑁 cor app 𝐸xc [𝑛1 , … , 𝑛𝑁 ] = 𝐸xc [𝑛] + ∑ 𝐸 SIC [𝑛𝑖 ] ,

(77)

𝑖=1

where 𝑛 is the sum of the one-orbital densities (cp. Eq. (65)). Here, the SIC energy of the 𝑖-th orbital is given by [24, Eq. 33] app 𝐸 SIC [𝑛𝑖 ] = −𝐸H [𝑛𝑖 ] − 𝐸xc [𝑛𝑖 ] .

(78)

The Perdew Zunger SIC can be applied to every spin-density exchange-correlation functional. Furthermore, it vanishes for the correct one (cp. Eq. (75)). For achieving a self-consistent theory, the exchange-correlation energy in the Hamiltonian SIC has to be replaced by 𝐸xc . The resulting Hamiltonian includes an orbital-dependent potential (cp. [24, Eqs. 37,38]). As the Perdew Zunger SIC includes this orbital dependent potential to exclude self-interaction (as Hartree-theory) as well as an approximation for exchange and correlation between different orbitals (as DFT), it can be seen as a combination of DFT and Hartree theory. It was shown, that Perdew Zunger SIC improves the results for exchangecorrelation holes compared to uncorrected L(S)DA calculations [24, Fig. 1]. As the Perdew Zunger SIC requires a correction for each orbital, it is computationally very demanding. In particular, the method is not size consistent, which means the correction does not scale consistently with size. Therefore, a Perdew Zunger SIC is in most cases not feasible. In 2014, M.R. Pederson et al. suggested a new method to correct the self-interaction error, that scales with 3𝑁 . Their method is based on localized orbitals and will be introduced in the next section.

2.3. Fermi-Löwdin Orbital Self-Interaction Correction This section gives an overview of the FLO SIC. For further information check [12, Ch. 8]. This method is based on three steps. Initially, the results from a preceding DFT calculation, the Kohn-Sham orbitals |𝜓𝛼 ⟩, are transferred to different orbitals, the Fermi orbitals |𝐹𝑖 ⟩. This transformation relies on an additional parameter, the Fermi orbital descriptor 𝒂𝑖 . This parameter is determined by minimizing the total energy. The thus defined Fermi orbitals are localized for the correct 𝒂𝑖 and therefore allegedly fulfill the localization equations. Unfortunately, the Fermi orbitals |𝐹𝑖 ⟩ are not orthonormal. For this reason, one secondly transforms the Fermi orbitals |𝐹𝑖 ⟩ to orthonormal Fermi Löwdin orbitals |𝜙𝑘 ⟩ by a symmetric Löwdin orthogonalization. Finally, one has to find the 𝒂𝑖 with minimal total energy. Since one introduces 𝑁 parameters whereof each contains three components, the FLO SIC scales with 3𝑁 . We will now consider these steps more detailed. Primarily, Pederson et al. have shown, that to achieve complete self-consistency, the local-


20


ization equations need to be satisfied [16]. Explicitly, they read [16, Eqs. 38, 39] 𝑁

̂ SIC ) |𝜙𝛼 ⟩ = ∑ 𝜆𝛽𝛼 |𝜙𝛽 ⟩ , (𝐻̂0 + 𝑉𝛼

(79)

𝛽=1

̂ SIC − 𝑉 ̂ SIC ∣𝜙𝛽 ⟩ = 0 , ⟨𝜙𝛼 ∣𝑉𝛼 𝛽

∀ 𝛼, 𝛽 .

(80)

Here, the orbital dependent SIC potential 𝑉𝛼SIC appears for the first time. It will be defined below. As explained by Perdew and Zunger, the main idea of FLO SIC is to correct the energy functional 𝐸0 [𝑛] by subtracting the SIC energy. Perdew and Zunger write this SIC energy as 𝑁

𝐸 SIC [𝑛1 , … , 𝑛𝑁 ] = ∑ 𝐸 SIC [𝑛𝑖 ] ,

(81)

𝑖=1

that has to be subtracted from the exchange correlation functional and therefore from the energy functional. Pederson et al. introduce a similar correction, where the SIC energy 𝐸 SIC is given via its derivative, the SIC potential 𝑣𝑖SIC , i.e. 𝛿 𝐸 SIC [𝑛𝑖 ] = 𝑣𝑖SIC [𝑛𝑖 ](𝒙) . 𝛿 𝑛𝑖 (𝒙)

(82)

The corresponding SIC potential is defined as (see [16, Eq. 10]) 𝑣𝑖SIC (𝒙) ∶= −

1/3 𝑒2 𝑛 (𝒙′ ) 𝑒2 3 ∫ d 3 𝒙′ 𝑖 − ( 𝑛 (𝒙)) . 4𝜋𝜀0 |𝒙 − 𝒙′ | 4𝜋𝜀0 𝜋 𝑖

(83)

It is related to the multiplicative SIC potential operator as (𝑉𝑖̂ SIC 𝜙𝑖 ) (𝒙) = 𝑣𝑖SIC (𝒙) 𝜙𝑖 (𝒙) .

(84)

The expectation value in a state |𝜙𝑖 ⟩ reads ⟨𝜙𝑖 ∣𝑉𝑖̂ SIC ∣𝜙𝑖 ⟩ = ∫ d3 𝒙 𝜙𝑖∗ (𝒙) (𝑉𝑖̂ SIC 𝜙𝑖 ) (𝒙) = ∫ d3 𝒙 𝜙𝑖∗ (𝒙) 𝑣𝑖SIC (𝒙) 𝜙𝑖 (𝒙) = ∫ d3 𝒙 𝑣𝑖SIC (𝒙) 𝑛𝑖 (𝒙) =− −

𝑛 (𝒙′ )𝑛𝑖 (𝒙) 𝑒2 ∫ d3 𝒙∫ d3 𝒙′ 𝑖 4𝜋𝜀0 |𝒙 − 𝒙′ | 𝑒2 3 1/3 4/3 ∫ d3 𝒙 𝑛𝑖 (𝒙) . ( ) 4𝜋𝜀0 𝜋

(85)

Here, the first term is the Hartree energy of one orbital |𝜙𝑖 ⟩, whereas the second term is its exchange correlation energy in LDA. Thus, the SIC potential operator as by Pederson follows the Perdew Zunger SIC. It is claimed that localized orbitals satisfy Eqs. (79) and (80) (cp. [16]). For this reason, the solution of the preceding DFT calculation, the Kohn-Sham orbitals, are transformed into localized orbitals. A special set of localized orbitals used in FLO SIC are Fermi orbitals.


2.3. Fermi-Löwdin Orbital Self-Interaction Correction

21

These Fermi orbitals |𝐹𝑖 ⟩ can be constructed from the Kohn-Sham orbitals |𝜓𝛼 ⟩ according to (see [21, Eq. 3]) 𝑁

|𝐹𝑖 ⟩ = ∑ 𝑅𝑖𝛼 |𝜓𝛼 ⟩ ,

(86)

𝛼=1

where

∗ 𝜓𝛼 (𝒂𝑖 )

𝑅𝑖𝛼 =

𝑁

.

(87)

2

√ ∑ |𝜓𝛼 (𝒂𝑖 )| 𝛼=1

The 𝒂𝑖 denote the respective Fermi orbital descriptors. They are determined by minimizing the SIC energy and hence the energy functional 𝐸[𝑛]. Unfortunately, the Fermi orbitals are not orthonormal, i.e. in general 𝑆𝑖𝑗 = ⟨𝐹𝑖 |𝐹𝑗 ⟩ ≠ 𝛿𝑖𝑗 .

(88)

Here, 𝑆𝑖𝑗 is the overlap matrix. Hence, it is reasonable to transform them to an orthonormal set of wave functions. This transformation is realized by a symmetric Löwdin orthogonalization. Therefore, the eigenvalues and eigenvectors of the overlap matrix are required. The corresponding eigenvalue problem reads 𝑁

∑ 𝑆𝑖𝑗 𝑇𝛼𝑗 = 𝑄𝛼 𝑇𝛼𝑖 .

(89)

𝑗=1

By solving this equation, we obtain the Löwdin eigenvalues 𝑄𝛼 and the matrix of Löwdin eigenvectors 𝑇𝛼𝑖 . With that, the transformation of the Fermi orbitals reads 𝑁

1 𝑇𝛼𝑘 𝑇𝛼𝑗 |𝐹𝑗 ⟩ . 𝛼,𝑗=1 √𝑄𝛼

|𝜙𝑘 ⟩ = ∑

(90)

Here, |𝜙𝑘 ⟩ are the Fermi Löwdin orbitals. They are now orthonormal, i.e. ⟨𝜙𝑙 |𝜙𝑘 ⟩ = 𝛿𝑙𝑘 .

(91)

The transformation between the Kohn-Sham orbitals and the Fermi Löwdin orbitals is unitary. Explicitly, this unitary transformation reads 𝑁

|𝜙𝑘 ⟩ =

1 𝑇𝛼𝑘 𝑇𝛼𝑗 𝑅𝑗𝑖 |𝜓𝑖 ⟩ . 𝛼,𝑖,𝑗=1 √𝑄𝛼 ∑

(92)

As a matter of principle, the ground state energy has to be minimized. The self-interaction corrected energy functional reads (cp. Eq. (77)) 𝐸 cor [𝑛1 , … , 𝑛𝑁 ] = 𝐸[𝑛] + 𝐸 SIC [𝑛1 , … , 𝑛𝑁 ] .

(93)

Hence, the minimization of the corrected energy functional includes the minimization of the SIC energy. Since the uncorrected energy functional has already been minimized in the preceding DFT calculation, the post proceeding FLO SIC sufficiently minimizes 𝐸 SIC . This minimization of the SIC energy is realized by a convergent gradient algorithm that requires the first order derivatives. Their derivation was done by Pederson [11, 12] and will be outlined in Sec. 3.1. As this minimization is computationally demanding, some acceleration is desirable. For this purpose, the second order derivatives of the SIC energy with respect to the Fermi orbital descriptors are useful (cp. Sec. A.1). Their derivation is the main purpose of this thesis. It is given in Secs. 3.2 and 3.3. : Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction

22

3. Derivatives

3. Derivatives This section introduces the main part of this thesis, i.e. the calculation of the second order derivatives of the SIC energy with respect to the Fermi orbital descriptors. We will first outline the derivation of the gradients, done by M. R. Pederson [11, 12] (see Secs. 3.1). Next, the second order derivatives will be derived (see Sec. 3.2) and rearranged for implementation (see Sec. 3.3).

3.1. First Derivative We will now derive the expressions for the first order derivative of the SIC energy with respect to the Fermi orbital descriptors. By an application of the chain rule for functional derivatives, this derivative can be written as 𝑁 𝑁 𝜕 𝐸 SIC [𝑛1 , … , 𝑛𝑁 ] 𝜕 𝐸 SIC [𝑛𝑖 ] 𝛿 𝐸 SIC [𝑛𝑖 ] 𝜕 𝑛𝑖 (𝒙) =∑ = ∑ ∫ d3 𝒙 . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝛿 𝑛𝑖 (𝒙) 𝜕 𝒂𝑚 𝑖=1 𝑖=1

(94)

With Eq. (82), this reads 𝑁 𝜕 𝐸 SIC 𝜕 𝑛𝑖 (𝒙) [𝑛 , … , 𝑛𝑁 ] = ∑ ∫ d3 𝒙 𝑣𝑖SIC [𝑛𝑖 ](𝒙) 𝜕 𝒂𝑚 1 𝜕 𝒂𝑚 𝑖=1 𝑁

= ∑ (− 𝑖=1

−

𝑒2 𝑛 (𝒙′ ) 𝜕 𝑛𝑖 (𝒙) ∫ d3 𝒙∫ d3 𝒙′ 𝑖 4𝜋𝜀0 |𝒙 − 𝒙′ | 𝜕 𝒂𝑚

𝜕 𝑛 (𝒙) 3 1/3 𝑒2 1/3 ( ) ∫ d3 𝒙 𝑛𝑖 (𝒙) 𝑖 ) . 4𝜋𝜀0 𝜋 𝜕 𝒂𝑚

(95)

Here, the one-orbital density 𝑛𝑖 is defined as 𝑛𝑖 (𝒙) = 𝜙𝑖∗ (𝒙) 𝜙𝑖 (𝒙) .

(96)

𝜕 𝑛𝑖 (𝒙) 𝜕 𝜙𝑖∗ (𝒙) 𝜕 𝜙 (𝒙) = 𝜙𝑖 (𝒙) + 𝜙𝑖∗ (𝒙) 𝑖 . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚

(97)

Its derivative is given by

With that, Eq. (95) reads 𝑁 𝜕 𝐸 SIC 𝜕 𝜙∗ (𝒙) 𝜕 𝜙 (𝒙) [𝑛1 , … , 𝑛𝑁 ] = ∑ ∫ d3 𝒙 𝑣𝑖SIC [𝑛𝑖 ](𝒙) ( 𝑖 𝜙𝑖 (𝒙) + 𝜙𝑖∗ (𝒙) 𝑖 ) . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑖=1

(98)

As is well known, the scalar product in the Dirac (bra-ket) notation is equal to an integration in the real space, i.e. (cp. Eq. (85)) ⟨𝜓 ∣ 𝐴 ̂ ∣ 𝜑⟩ = ∫ d3 𝒙 𝜓∗ (𝒙) (𝐴 ̂ 𝜑)(𝒙) .

(99)

With this and with Eq. (84), the derivative of the SIC energy can be written as 𝑁 𝜕 𝐸 SIC 𝜕 𝜙𝑖 𝜕 𝜙𝑖∗ (𝒙) ̂ SIC = ∑ (∫ d3 𝒙 (𝑉𝑖 𝜙𝑖 ) (𝒙) + ∫ d3 𝒙 𝜙𝑖∗ (𝒙) (𝑉𝑖̂ SIC ) (𝒙)) 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑖=1 𝑁

= ∑ (⟨ 𝑖=1

𝜕 𝜙𝑖 𝜕 𝜙𝑖 ∣ 𝑉 ̂ SIC ∣ 𝜙𝑖 ⟩ + ⟨𝜙𝑖 ∣ 𝑉𝑖̂ SIC ∣ ⟩) . 𝜕 𝒂𝑚 𝑖 𝜕 𝒂𝑚

(100)


3.1. First Derivative

23

The following derivation follows [11, 12], where Pederson rearranged the derivative of the SIC energy to simplify its implementation into the current NRLMOL computer code. The derivation by Pederson starts with the above bra-ket notation [12, Eq. 8.16]. Since the SIC energy has to be a real value, its derivative and therefore the sum of all inner products in Eq. (100) have to be real as well. Evidently they are, since we add only terms and their complex conjugate counter parts. The first derivative can hence be written as 𝑁 𝜕 𝐸 SIC 𝜕 𝜙𝑘 = 2 ∑ ℛ𝑒 (⟨𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑘=1

(101)

For this reason, we will only consider real inner products in the following. If all inner products are real, one can transpose and reorder them without any effect. We now include the projector 𝑃 ̂ on the occupied subspace, i.e. 𝑁

𝑃 ̂ = ∑ |𝜙𝑙 ⟩ ⟨𝜙𝑙 | .

(102)

𝑙=1

Since the derivative of the Fermi Löwdin orbitals with respect to the Fermi orbital descriptors is still an element of the occupied subspace, an application of this projector to Eq. (100) is equivalent to the insertion of an identity operator, i.e. it does not change the derivative. An insertion of the identity operator and the former assumption lead to the following restatements of Eq. (100), 𝑁 𝜕 𝐸 SIC 𝜕 𝜙𝑘 𝜕 𝜙𝑘 = ∑ (⟨ ∣ 𝜙𝑙 ⟩ ⟨𝜙𝑙 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑘 ⟩ + ⟨𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ ⟨𝜙𝑙 ∣ ⟩) , 𝜕 𝒂𝑚 𝜕 𝒂 𝜕 𝒂𝑚 𝑚 𝑘,𝑙=1 𝑁 𝜕 𝜙𝑘 𝜕 𝜙𝑘 = ∑ ⟨𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ (⟨ ∣ 𝜙𝑙 ⟩ + ⟨𝜙𝑙 ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑘,𝑙=1

(103)

We now define ⟨𝜙𝑗 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ ≡ 𝜖𝑘 𝑗𝑙 ,

(104)

𝑁 𝜕 𝐸 SIC 𝜕 𝜙𝑘 𝜕 𝜙𝑘 = ∑ 𝜖𝑘 ∣ 𝜙𝑙 ⟩ + ⟨𝜙𝑙 ∣ ⟩) . 𝑘𝑙 (⟨ 𝜕 𝒂𝑚 𝜕 𝒂 𝜕 𝒂𝑚 𝑚 𝑘,𝑙=1

(105)

and rewrite Eq. (103) as

Next, we add zero, which reads 𝑁 ⎛ ⎞ 𝜕 𝐸 SIC ⎜ ⎜⟨ 𝜕 𝜙𝑘 ∣ 𝜙𝑙 ⟩ + ⟨𝜙𝑙 ∣ 𝜕 𝜙𝑘 ⟩ + ⟨ 𝜕 𝜙𝑙 ∣ 𝜙𝑘 ⟩ − ⟨ 𝜕 𝜙𝑙 ∣ 𝜙𝑘 ⟩⎟ ⎟ = ∑ 𝜖𝑘 . ⎟ 𝑘𝑙 ⎜ ⎜ 𝜕 𝒂𝑚 ⎟ 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑘,𝑙=1 ⎝ ⎠ =0 (106) The Fermi Löwdin orbitals are orthonormal. As a consequence, the derivative of their scalar product vanishes, i.e.

𝜕 𝜕 𝛿𝑙𝑘 𝜕 𝜙𝑘 𝜕 𝜙𝑙 ⟨𝜙𝑙 | 𝜙𝑘 ⟩ = = ⟨𝜙𝑙 ∣ ⟩+⟨ ∣𝜙 ⟩ = 0 . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑘


(107)

24

3. Derivatives

With that, Eq. (106) reads 𝑁 𝜕 𝐸 SIC 𝜕 𝜙𝑙 𝜕 𝜙𝑘 = ∑ 𝜖𝑘 ∣ 𝜙𝑙 ⟩ − ⟨ ∣ 𝜙𝑘 ⟩) ≡ ∑ 𝜖𝑘 𝑘𝑙 (⟨ 𝑘𝑙 Δ𝑙𝑘,𝑚 . 𝜕 𝒂𝑚 𝜕 𝒂 𝜕 𝒂 𝑚 𝑚 𝑘,𝑙=1 𝑘

(108)

The Fermi Löwdin orbitals are constructed from the Fermi orbitals according to Eq. (90). Since all three of 𝑄𝛼 , 𝑇𝛼𝑖 and |𝐹𝑖 ⟩ depend on the Fermi orbital descriptors 𝒂𝑚 , the derivative reduces to 𝜕 𝜙𝑘 ⟩ = |𝐷1,𝑘𝑚 ⟩ + |𝐷2,𝑘𝑚 ⟩ + |𝐷3,𝑘𝑚 ⟩ , (109) ∣ 𝜕 𝒂𝑚 where 𝑁

𝜕 𝐹𝑗 1 ⟩ , 𝑇𝛼𝑘 𝑇𝛼𝑗 ∣ 𝜕 𝒂𝑚 𝛼,𝑗=1 √𝑄𝛼

|𝐷1,𝑘𝑚 ⟩ = ∑ |𝐷2,𝑘𝑚 ⟩ = −

(110)

𝜕 𝑄𝛼 1 1 𝑁 ∑ 𝑇𝛼𝑘 𝑇𝛼𝑗 |𝐹 ⟩ , 3/2 2 𝛼,𝑗=1 𝑄𝛼 𝜕 𝒂𝑚 𝑗

(111)

𝑁

𝜕 𝑇𝛼𝑗 𝜕𝑇 1 ( 𝛼𝑘 𝑇𝛼𝑗 + 𝑇𝛼𝑘 ) |𝐹𝑗 ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝛼,𝑗=1 √𝑄𝛼

|𝐷3,𝑘𝑚 ⟩ = ∑

(112)

In the following, we use the notation of [11, 12]. Since the difference of the brakets ⟨𝜙𝑙 | 𝜕 𝜙𝑘 /𝜕 𝒂𝑚 ⟩ and ⟨𝜙𝑘 | 𝜕 𝜙𝑙 /𝜕 𝒂𝑚 ⟩ is required for the first derivative of the SIC energy and since the derivative of the Fermi Löwdin orbitals is given as the sum of |𝐷1,𝑘𝑚 ⟩, |𝐷2,𝑘𝑚 ⟩ and |𝐷3,𝑘𝑚 ⟩, the subtraction and the addition can be interchanged, whereby the expression Δ𝑖𝑙𝑘,𝑚 = ⟨𝜙𝑙 | 𝐷𝑖,𝑘𝑚 ⟩ − ⟨𝜙𝑘 | 𝐷𝑖,𝑙𝑚 ⟩

(113)

for 𝑖 = 1, 2, 3 makes sence. We will therefore not only consider the terms of Eq. (109), but also these differences. For the convenience of the reader, we consider the three terms separately. We start with the first term |𝐷1,𝑘𝑚 ⟩. It is evident, that each Fermi orbital |𝐹𝑖 ⟩ only depends on its own Fermi orbital descriptor 𝒂𝑖 . From this it follows that ∣

𝜕 𝐹𝑖 𝜕 𝐹𝑚 ⟩ = 𝛿𝑖𝑚 ∣ ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑚

(114)

With this, Eq. (110) simplifies to 𝑁

1 𝜕 𝐹𝑚 𝑇𝛼𝑘 𝑇𝛼𝑚 ∣ ⟩ . 𝜕 𝒂𝑚 𝛼=1 √𝑄𝛼

|𝐷1,𝑘𝑚 ⟩ = ∑

(115)

Its scalar product with a Fermi Löwdin orbital |𝜙𝑙 ⟩ is given by 𝑁

⟨𝜙𝑙 | 𝐷1,𝑘𝑚 ⟩ =

1 1 𝜕 𝐹𝑚 𝑇𝛼𝑘 𝑇𝛼𝑚 𝑇𝛽𝑙 𝑇𝛽𝑛 ⟨𝐹𝑛 ∣ ⟩ . 𝜕 𝒂𝑚 √𝑄𝛽 𝛼,𝛽,𝑛=1 √𝑄𝛼 ∑

(116)

Therefore, the difference Δ1𝑙𝑘,𝑚 reduces to 𝑁

Δ1𝑙𝑘,𝑚 =

∑ 𝛼,𝛽,𝑛=1

(𝑇𝛼𝑘 𝑇𝛽𝑙 − 𝑇𝛼𝑙 𝑇𝛽𝑘 ) 𝑇𝛼𝑚 𝑇𝛽𝑛 √𝑄𝛼 𝑄𝛽

⟨𝐹𝑛 ∣

𝜕 𝐹𝑚 ⟩ . 𝜕 𝒂𝑚

(117)


3.1. First Derivative

25

We now turn to ⟨𝜙𝑟 | 𝐷2,𝑘𝑚 ⟩. From Eq. (111), it follows that ⟨𝜙𝑟 | 𝐷2,𝑘𝑚 ⟩ = −

𝑁 1 1 𝜕 𝑄𝛼 1 ∑ 𝑇𝛼𝑘 𝑇𝛼𝑗 𝑇 𝑇 𝑆 . 3/2 2 𝛼,𝛽,𝑛,𝑗=1 𝑄𝛼 𝜕 𝒂𝑚 √𝑄𝛽 𝛽𝑙𝑟 𝛽𝑛 𝑛𝑗

(118)

Hence, an expression for the derivative of the Löwdin eigenvalues with respect to the Fermi orbital descriptors is required. For this purpose, the perturbation analysis is useful. From Eq. (89) it follows that 𝑁

𝑁

𝑁

∑ 𝑇𝛼𝑖 ∑ 𝑆𝑖𝑗 𝑇𝛼𝑗 = ∑ 𝑇𝛼𝑖 𝑄𝛼 𝑇𝛼𝑖 . 𝑗=1

𝑖=1

(119)

𝑖=1

The orthonormality of the Löwdin eigenvectors (see Eq. (91)) implies that 𝑁

∑ 𝑇𝛼𝑖 𝑇𝛼𝑖 = 𝛿𝛼𝛼 ≡ 1 ,

(120)

𝑖=1

and Eq. (119) transforms to 𝑁

𝑄𝛼 = ∑ 𝑇𝛼𝑖 𝑆𝑖𝑗 𝑇𝛼𝑗 .

(121)

𝑖,𝑗=1

By a straightforward transformation, we obtain an expression for the derivative. Explicitly, it reads 𝑁 𝜕 𝑄𝛼 𝜕 = (∑ 𝑇 𝑆 𝑇 ) 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑖,𝑗=1 𝛼𝑖 𝑖𝑗 𝛼𝑗 𝑁 𝑁 𝜕 𝑆𝑖𝑗 𝜕 𝑇𝛼𝑗 𝜕 𝑇𝛼𝑖 𝑆𝑖𝑗 𝑇𝛼𝑗 + ∑ 𝑇𝛼𝑖 𝑇𝛼𝑗 + ∑ 𝑇𝛼𝑖 𝑆𝑖𝑗 . 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑖,𝑗=1 𝑖,𝑗=1 𝑖,𝑗=1 𝑁

= ∑

(122)

To calculate the derivative of the Löwdin eigenvectors with respect to the Fermi orbital descriptors we introduce the intermediate Löwdin orbitals, that read [12, Ch. 8, Eq. 10] 𝑁

|𝑇𝛼 ⟩ = ∑ 𝑇𝛼𝑗 |𝐹𝑗 ⟩ .

(123)

𝑗=1

We now apply a first-order perturbation analysis to these orbitals, which yields

(1)

|𝑇𝛼 ⟩

𝑁

= |𝑇𝛼 ⟩ + ∣

𝑁

𝜕 𝑇𝛼 ⟩ Δ𝒂𝑚 = ∑ 𝑇𝛼𝑗 |𝐹𝑗 ⟩ + ∑ |𝑇𝛽 ⟩ 𝜕 𝒂𝑚 𝑗=1 𝛼,𝛽=1

d𝑆 ∣𝑇 ⟩ d 𝒂𝑚 𝛼 Δ𝒂𝑚 , 𝑄𝛼 − 𝑄 𝛽

⟨𝑇𝛽 ∣

𝛼≠𝛽 𝑁

𝑁

𝜕 𝑇𝛼𝑗 |𝐹 ⟩ Δ𝒂𝑚 . 𝜕 𝒂𝑚 𝑗 𝑗=1

= ∑ 𝑇𝛼𝑗 |𝐹𝑗 ⟩ + ∑ 𝑗=1

(124)

By equating the coefficients it follows that 𝑁

𝑁 𝜕 𝑇𝛼𝑗 ∑ |𝐹 ⟩ = ∑ |𝑇𝛽 ⟩ 𝜕 𝒂𝑚 𝑗 𝑗=1 𝛼,𝛽=1 𝛼≠𝛽

d𝑆 d𝑆 ⟨𝑇𝛽 ∣ ∣𝑇 ⟩ ∣𝑇 ⟩ 𝑁 𝑁 d 𝒂𝑚 𝛼 d 𝒂𝑚 𝛼 = ∑ ∑ 𝑇𝛽𝑗 |𝐹𝑗 ⟩ , 𝑄𝛼 − 𝑄 𝛽 𝑄𝛼 − 𝑄 𝛽 𝛼,𝛽=1 𝑗=1

⟨𝑇𝛽 ∣

𝛼≠𝛽

(125) : Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction

26

3. Derivatives

which results in 𝑁

𝜕 𝑇𝛼𝑗 = ∑ 𝑇𝛽𝑗 𝜕 𝒂𝑚 𝛼,𝛽=1

d𝑆 ∣𝑇 ⟩ d 𝒂𝑚 𝛼 . 𝑄𝛼 − 𝑄 𝛽

⟨𝑇𝛽 ∣

(126)

𝛼≠𝛽

With that, the derivative of the Löwdin eigenvalues with respect to the Fermi orbital descriptors is given. Since the columns of 𝑇𝛼𝑖 are eigenvectors of 𝑆𝑖𝑗 , the dot product of two of them has to be orthonormal. Straightforwardly, we conclude that ∑ 𝑇𝛽𝑗 𝑇𝛼𝑗 = 𝛿𝛼𝛽 .

(127)

𝑗

As a consequence, Δ2𝑙𝑘,𝑚 disappears. Further information and a detailed derivation are given in [12, Ch. 8, Sec. 3.2]. We now consider the third term of Eq. (109). With Eq. (126) We have obtained an expression for the required derivative of the eigenvectors 𝑇𝛼𝑖 with respect to the Fermi orbital descriptors 𝒂𝑚 . We emphasize, that the perturbation analysis assumes only small perturbations. As the minimization algorithm should go sufficiently small steps, especially close to the minimum, we consider this condition as accomplished. We now insert this result in Eqs. (112) and (113). We therefore rewrite ⟨𝜙𝑙 | 𝐷3,𝑘𝑚 ⟩ and Δ3𝑙𝑘,𝑚 as 𝑁

⟨𝜙𝑙 | 𝐷3,𝑘𝑚 ⟩ = −

1 ∑ 2 𝛼,𝛽=1 𝑁

Δ3𝑙𝑘,𝑚 = −

1 ∑ 2 𝛼,𝛽=1

⟨𝑇𝛽 ∣

d𝑆 1/2 1/2 ∣ 𝑇 ⟩ (𝑇𝛽𝑘 𝑇𝛼𝑙 𝑄𝛼 + 𝑇𝛼𝑘 𝑇𝛽𝑙 𝑄𝛽 ) d 𝒂𝑚 𝛼 1/2

1/2

1/2

,

(128)

(𝑄𝛼 + 𝑄𝛽 ) (𝑄𝛼 𝑄𝛽 ) ⟨𝑇𝛽 ∣

d𝑆 1/2 1/2 ∣ 𝑇 ⟩ (𝑇𝛼𝑘 𝑇𝛽𝑙 − 𝑇𝛼𝑙 𝑇𝛽𝑘 ) (𝑄𝛽 − 𝑄𝛼 ) d 𝒂𝑚 𝛼 1/2

1/2

(𝑄𝛼 + 𝑄𝛽 ) (𝑄𝛼 𝑄𝛽 )

1/2

.

(129)

Since Δ2𝑙𝑘,𝑚 disappears, the heretofore undefined part Δ𝑙𝑘,𝑚 of Eq. (108) simplifies to Δ𝑙𝑘,𝑚 = Δ1𝑙𝑘,𝑚 + Δ3𝑙𝑘,𝑚 .

(130)

These analytical expressions for the first derivative are implemented in the recent computer code of NRLMOL. The minimization of the SIC energy is realized by a convergent gradient algorithm.

3.2. Second Order Derivative In this section, we will derive the second order derivative of the SIC energy with respect to the Fermi orbital descriptors. Starting from Eq. (94), the Hessian of the SIC energy is given by 𝑁 𝑁 𝜕 2 𝐸 SIC 𝜕 2 𝐸 SIC 𝜕 𝛿 𝐸 SIC [𝑛𝑖 ] 𝜕 𝑛𝑖 (𝒙) =∑ =∑ (∫ d3 𝒙 ) . (131) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑛 𝛿 𝑛𝑖 (𝒙) 𝜕 𝒂𝑚 𝑖=1 𝑖=1 Fundamentally, the application of the chain rule for functional derivatives yields 𝜕 2 𝐸 SIC [𝑛𝑖 ] 𝛿 2 𝐸 SIC [𝑛𝑖 ] 𝜕 𝑛𝑖 (𝒙) 𝜕 𝑛𝑖 (𝒙′ ) = ∫ d3 𝒙∫ d3 𝒙′ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝛿 𝑛𝑖 (𝒙) 𝛿 𝑛𝑖 (𝒙′ ) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝛿 𝐸 SIC [𝑛𝑖 ] 𝜕 2 𝑛𝑖 (𝒙) + ∫ d3 𝒙 . 𝛿 𝑛𝑖 (𝒙) 𝜕 𝒂𝑚 𝜕 𝒂𝑛

(132)


3.2. Second Order Derivative

27

Using the definition of the SIC potential given in Eq. (82), the derivative simplifies to 𝜕 2 𝐸 SIC [𝑛𝑖 ] 𝛿 𝑣SIC [𝑛𝑖 ](𝒙) 𝜕 𝑛𝑖 (𝒙) 𝜕 𝑛𝑖 (𝒙′ ) = ∫ d3 𝒙∫ d3 𝒙′ 𝑖 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝛿 𝑛𝑖 (𝒙′ ) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 2 𝜕 𝑛𝑖 (𝒙) + ∫ d3 𝒙 𝑣𝑖SIC (𝒙)[𝑛𝑖 ] . 𝜕 𝒂𝑚 𝜕 𝒂 𝑛

(133)

The differentiation of the SIC potential with respect to the density yields 𝛿 𝑣𝑖SIC (𝒙)[𝑛𝑖 ] 𝛿 2 𝐸 SIC [𝑛𝑖 ] = 𝛿 𝑛𝑖 (𝒙′ ) 𝛿 𝑛𝑖 (𝒙) 𝛿 𝑛𝑖 (𝒙′ ) =−

1 3 1/3 1 −2/3 𝑒2 𝑒2 − ( ) 𝑛 (𝒙) 𝛿 3 (𝒙 − 𝒙′ ) . 4𝜋𝜀0 |𝒙 − 𝒙′ | 4𝜋𝜀0 𝜋 3 𝑖

(134)

With that, the Hessian of the SIC energy reads 𝜕 2 𝑛𝑖 (𝒙) 𝜕 2 𝐸 SIC [𝑛𝑖 ] 𝜕 𝑛𝑖 (𝒙′ ) 𝜕 𝑛 (𝒙) = ∫ d3 𝒙 𝑣𝑖SIC (𝒙) − 𝑒2 ∫ d3 𝒙∫ d3 𝒙′ 𝑣(𝒙, 𝒙′ ) 𝑖 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂𝑛 𝜕 𝒂𝑚 −

3 1/3 𝜕 𝑛𝑖 (𝒙) 𝜕 𝑛𝑖 (𝒙) −2/3 1 𝑒2 (𝒙) . ( ) ∫ d3 𝒙 𝑛 3 4𝜋𝜀0 𝜋 𝜕 𝒂𝑛 𝜕 𝒂 𝑚 𝑖

(135)

This is the expression for the second order derivative of the SIC energy with respect to the Fermi orbital descriptors. We rearranged this derivative into terms of the derivatives of the density. The first derivative has already been defined in Eq. (97). The second order derivative of the density with respect to the Fermi orbital descriptors reads in the position state basis 𝜕 2 𝜙𝑖∗ (𝒙) 𝜕 𝜙𝑖∗ (𝒙) 𝜕 𝜙𝑖 (𝒙) 𝜕 2 𝑛𝑖 (𝒙) = 𝜙𝑖 (𝒙) + 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑛 𝜕 𝒂 𝑚 ∗ 𝜕 𝜙𝑖 (𝒙) 𝜕 𝜙𝑖 (𝒙) 𝜕 2 𝜙𝑖 (𝒙) + + 𝜙𝑖∗ (𝒙) . 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂 𝑛

(136)

Hence, the derivative of the Fermi Löwdin orbital with respect to the Fermi orbital descriptors is required. We already know its expression, but not in the real space. Eq. (109) yields ∣

𝜕 𝜙𝑘 ⟩ = |𝐷1,𝑘𝑛 ⟩ + |𝐷2,𝑘𝑛 ⟩ + |𝐷3,𝑘𝑛 ⟩ . 𝜕 𝒂𝑛

(137)

In the position state basis, this reads 𝜕 𝜙𝑘 (𝒙) 𝜕 𝜙𝑘 = ⟨𝒙 ∣ ⟩ . 𝜕 𝒂𝑛 𝜕 𝒂𝑛

(138)

The derivatives of the Fermi Löwdin orbitals are implemented as ⟨𝜙𝑙 ∣

𝜕 𝜙𝑘 ⟩ = ⟨𝜙𝑙 ∣ 𝐷1,𝑘𝑛 ⟩ + ⟨𝜙𝑙 ∣ 𝐷2,𝑘𝑛 ⟩ + ⟨𝜙𝑙 ∣ 𝐷3,𝑘𝑛 ⟩ . 𝜕 𝒂𝑛

(139)

With that, the derivative of the Fermi Löwdin orbitals in the real space reads 𝑁 𝜕 𝜙𝑘 (𝒙) 𝜕 𝜙𝑘 = ∑ ⟨𝒙 | 𝜙𝑙 ⟩ ⟨𝜙𝑙 ∣ ⟩ . 𝜕 𝒂𝑛 𝜕 𝒂𝑛 𝑙=1


(140)

28

3. Derivatives

An expression for the second order derivative of |𝑝ℎ𝑖𝑘 ⟩ with respect to the Fermi orbital descriptors is still required. The implementation of a new expressions in the NRLMOL computer code is most straightforward when done analogously to already implemented expressions. Since the first order derivatives are implemented in analogy to the bra-ket noted expressions, it would be useful to rearrange the second order derivatives into expressions, which do only depend on terms of the first order derivative. We note, that the second order derivative of the SIC energy does not only depend on terms of the first derivative of the density. Nonetheless, we will rearrange the bra-ket noted version of the second order derivative. For this purpose, we plug Eq. (136) into Eq. (135), which yields 𝜕 2 𝐸 SIC 𝜕 2 𝜙𝑖∗ (𝒙) SIC = ∫ d3 𝒙 𝑣 (𝒙)𝜙𝑖 (𝒙) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝑖 𝜕 𝜙 (𝒙) 𝜕 𝜙𝑖∗ (𝒙) SIC + ∫ d3 𝒙 𝑣𝑖 (𝒙) 𝑖 𝜕 𝒂𝑛 𝜕 𝒂𝑚 ∗ 𝜕 𝜙 (𝒙) 𝜕 𝜙𝑖 𝑖 + ∫ d3 𝒙 𝑣𝑖SIC (𝒙) (𝒙) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 2 𝜙𝑖 (𝒙) + ∫ d3 𝒙 𝜙𝑖∗ (𝒙)𝑣𝑖SIC (𝒙) 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝑛𝑖 (𝒙′ ) 𝜕 𝑛 (𝒙) − 𝑒2 ∫ d3 𝒙∫ d3 𝒙′ 𝑣(𝒙, 𝒙′ ) 𝑖 𝜕 𝒂𝑛 𝜕 𝒂𝑚 −

3 1/3 𝜕 𝑛𝑖 (𝒙) 𝜕 𝑛𝑖 (𝒙) −2/3 1 𝑒2 (𝒙) . ( ) ∫ d3 𝒙 𝑛 3 4𝜋𝜀0 𝜋 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝑖

(141)

With that, a good starting point for the rearrangement in the next section is given.

3.3. Rearrangement of the Hessian We will now rearrange the Hessian into a form similar to the first derivative in the bra-ket notation. First, we rewrite the last two terms of Eq. (141) as ∫ d3 𝒙

𝜕 𝑣𝑖SIC (𝒙) 𝜕 𝑛𝑖 (𝒙) 𝜕 𝜙𝑖 𝜕 𝑉𝑖̂ SIC 𝜕 𝑉𝑖̂ SIC 𝜕 𝜙𝑖 =⟨ ∣ ∣ 𝜙𝑖 ⟩ + ⟨𝜙𝑖 ∣ ∣ ⟩ , 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂 𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑛 𝜕 𝒂𝑚

(142)

because the derivative of the SIC potential with respect to the Fermi orbital descriptors reads (cp. Eq. (134)) 𝜕 𝑣𝑖SIC 𝛿 𝑣𝑖SIC 𝜕 𝑛𝑖 (𝒙) = ∫ d3 𝒙 𝜕 𝒂𝑛 𝛿 𝑛𝑖 (𝒙) 𝜕 𝒂𝑛 = −𝑒2 ∫ d3 𝒙′ 𝑣(𝒙, 𝒙′ )

𝜕 𝑛 (𝒙) 𝜕 𝑛𝑖 (𝒙′ ) 𝑒2 1 3 1/3 −2/3 ( ) 𝑛𝑖 (𝒙) 𝑖 − . 𝜕 𝒂𝑛 4𝜋𝜀0 3 𝜋 𝜕 𝒂𝑛

(143)

We hence rewrite Eq. (141) analogously to Eq. (100) as 𝑁 𝜕 2 𝐸 SIC 𝜕 2 𝜙𝑖 𝜕 𝜙𝑖 𝜕 𝜙𝑖 𝜕 2 𝜙𝑖 = ∑ (⟨ ∣ 𝑉𝑖̂ SIC ∣ 𝜙𝑖 ⟩ + 2 ⟨ ∣ 𝑉𝑖̂ SIC ∣ ⟩ + ⟨𝜙𝑖 ∣ 𝑉𝑖̂ SIC ∣ ⟩ 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑖=1

+⟨

𝜕 𝑉𝑖̂ SIC 𝜕 𝜙𝑖 𝜕 𝜙𝑖 𝜕 𝑉𝑖̂ SIC ∣ ∣ 𝜙𝑖 ⟩ + ⟨𝜙𝑖 ∣ ∣ ⟩) 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂 𝑛 𝜕 𝒂𝑚

= 𝐴(𝒂𝑛 , 𝒂𝑚 ) + 𝐵(𝒂𝑛 , 𝒂𝑚 ) .

(144)


3.3. Rearrangement of the Hessian

29

where 𝐴 and 𝐵 are given by 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = ∑ (⟨ 𝑘=1

𝜕 2 𝜙𝑘 𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑘 ⟩ + 2 ⟨ ∣ 𝑉𝑘̂ SIC ∣ ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂𝑛 𝜕 𝒂𝑚

+ ⟨𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ 𝑁

𝐵(𝒂𝑛 , 𝒂𝑚 ) = ∑ (⟨ 𝑘=1

𝜕 2 𝜙𝑘 ⟩) , 𝜕 𝒂𝑚 𝜕 𝒂𝑛

𝜕 𝑉𝑘̂ SIC 𝜕 𝜙𝑘 𝜕 𝜙𝑘 𝜕 𝑉𝑘̂ SIC ∣ ∣ 𝜙𝑘 ⟩ + ⟨𝜙𝑘 ∣ ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂𝑛 𝜕 𝒂𝑚

(145) (146)

The first term of the second order derivative, 𝐴, depends on second order derivatives of the Fermi Löwdin orbitals |𝜙𝑘 ⟩ and will be transformed in the following. The second term of the second order derivative 𝐵 is equal to the first order derivative of the auxiliary energy aux 𝐸𝑘aux , whose derivative with respect to 𝑛 is given by 𝑣𝑘 , which is the derivative of the SIC potential. Explicitly, 𝐵 reads 𝑁

𝐵(𝒂𝑛 , 𝒂𝑚 ) = ∑ (⟨ 𝑘=1

𝑁 𝜕 𝐸𝑘aux 𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝑉𝑘̂ aux ∣ 𝜙𝑘 ⟩ + ⟨𝜙𝑘 ∣ 𝑉𝑘̂ aux ∣ ⟩) = ∑ , 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑘=1

with

SIC 𝜕 𝑣𝑘 (𝒙) 𝛿 𝐸𝑘aux aux = 𝑣𝑘 (𝒙) = . 𝛿 𝑛𝑖 (𝒙) 𝜕 𝒂𝑛

(147)

(148)

Apparently, 𝐵 is already given in terms of the first order derivative of 𝐸𝑘SIC if we replace the operator 𝑉𝑘̂ SIC with its derivative 𝑉𝑘̂ aux . Therefore, we do not need to transform it. We will now try to transform 𝐴 into terms of the first order derivative. As in the derivation of the first order derivative, we only consider real inner products, since we add again a complex term and its complex conjugate. The expectation value of 𝑉𝑘̂ SIC in the derivative of the Fermi Löwdin orbitals has to be real, since 𝑉𝑘̂ SIC is Hermitian. We can hence write the first term of the second order derivative as 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = ∑ (2 ℛ𝑒 (⟨ 𝑘=1

𝜕 𝜙𝑘 𝜕 𝜙𝑘 𝜕 2 𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑘 ⟩) + 2 ⟨ ∣ 𝑉𝑘̂ SIC ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝜕 𝒂𝑚 𝜕 𝒂𝑛

(149)

Therefore, all inner products in 𝐴(𝒂𝑛 , 𝒂𝑚 ) are assumed to be real and one hence can transpose them and change their order without modifying the derivative. We will omit the ℛ𝑒 in the following transformations. We now insert the projector on the occupied subspace 𝑃 ̂ (cp. Eq. (102)). With that, the preceding expression yields 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) =

∑ (2 ⟨ 𝑗,𝑘,𝑙=1

𝜕 2 𝜙𝑘 ∣ 𝜙 ⟩ ⟨𝜙𝑗 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ ⟨𝜙𝑙 | 𝜙𝑘 ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑗

+2⟨

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑗 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ ⟨𝜙𝑙 ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂𝑛

(150)

Now we place ⟨𝜙𝑗 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ outside the brackets, which yields 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = 2 ∑ ⟨𝜙𝑗 ∣ 𝑉𝑘̂ SIC ∣ 𝜙𝑙 ⟩ (⟨ 𝑗,𝑘,𝑙=1

𝜕 2 𝜙𝑘 ∣ 𝜙 ⟩ ⟨𝜙𝑙 | 𝜙𝑘 ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑗

+⟨

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑙 ∣ ⟩) . 𝜕 𝒂𝑚 𝜕 𝒂𝑛


(151)

30

4. Conclusion

Analogously to Eq. (104), we define ̂ SIC ∣ 𝜙𝑙 ⟩ , 𝜆𝑘 𝑗𝑙 ∶= ⟨𝜙𝑗 ∣ 𝑉𝑘

(152)

and hence rewrite the Hessian according to 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = 2 ∑ 𝜆𝑘 𝑗𝑙 (⟨ 𝑗,𝑘,𝑙=1

𝜕 2 𝜙𝑘 ∣ 𝜙 ⟩ ⟨𝜙𝑙 | 𝜙𝑘 ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑗

+⟨

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙 ⟩ ⟨𝜙𝑙 ∣ ⟩) . 𝜕 𝒂𝑚 𝑗 𝜕 𝒂𝑛

(153)

Due to the orthonormality of the Fermi Löwdin orbitals (cp. Eq. (91)), the sum reduces to 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = 2 ∑ 𝜆𝑘 𝑗𝑘 ⟨ 𝑗,𝑘=1

𝜕 2 𝜙𝑘 ∣𝜙 ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑗

𝑁

+ 2 ∑ 𝜆𝑘 𝑗𝑙 ⟨ 𝑗,𝑘,𝑙=1

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑙 ∣ ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑛

(154)

In terms of the first order derivative, the second term of Eq. (154) reads ⟨

3 𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑙 ∣ ⟩ = ∑ ⟨𝐷ℎ,𝑘𝑚 | 𝜙𝑗 ⟩ ⟨𝜙𝑙 | 𝐷𝑖,𝑘𝑛 ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑛 ℎ,𝑖=1

(155)

The matrix elements ⟨𝜙𝑘 | 𝐷𝑖,𝑙𝑚 ⟩ for 𝑖 = 1, 2, 3 are defined in Eqs. (116),(118) and (128). We remember that the intention of the preceding transformations was to rewrite the second order derivatives in terms of their first order counter parts. As noted above, this is possible for the second expression appearing in Eq. (154), B. Unfortunately, it is not possible for the first expression of Eq. (154), A. Thus, the derivative of Eq. (109) with respect to the Fermi orbital descriptors is still required. Since each term of |𝐷1,𝑘𝑚 ⟩, |𝐷2,𝑘𝑚 ⟩ and |𝐷3,𝑘𝑚 ⟩ depends on the Fermi orbital descriptors, the second order derivative is very expensive and would lie outside the scope of this document. Nonetheless, some ideas how the second order derivatives can be useful for the minimization of the SIC energy are given in Sec. A.1.

4. Conclusion In this section, we will give a short summary of the present document. Moreover, we will give some ideas, how the future of FLO SIC might look like. This document reviews the origin of the self-interaction error and its elimination with SIC by J. P. Perdew and A. Zunger as well as by M. R. Pederson. The derivation and rearrangement of the first of the SIC energy with respect to the Fermi orbital descriptors are outlined and the derivative of the Löwdin eigenvalues with respect to the Fermi orbital descriptors has been derived. Furthermore, the derivation of the second order derivative is given. This has been the main purpose of this thesis. In addition, the Appendix gives some ideas, how to use second order derivatives in numerical minimization algorithms and summarizes the main expressions of the derivatives. We suggest to implement the second order derivative as given in Eq. (141). Alternatively, an implementation using Eq. (154) would be possible. In both cases, the second order derivative of the Fermi Löwdin orbitals with respect to the Fermi orbital descriptors is required. We : Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction

31

expect that the usage of the Hessian of the SIC energy as a Preconditioner will accelerate its minimization. Moreover, the expressions derived in this work for the derivative of the Löwdin eigenvalues with respect to the Fermi orbital descriptors allow to avoid the minimization of the SIC energy. For physical reasons, we assume that each of the Fermi orbitals with the correct Fermi orbital descriptors contains exactly one electron, i.e. the corresponding Löwdin eigenvalues are all equal to one. Hence, the optimization of the Fermi orbital descriptors via the minimization of the SIC energy could be replaced by a minimization of the difference from the Löwdin eigenvalues to one. This process requires the first order derivatives of the Löwdin eigenvalues with respect to the Fermi orbital descriptors. Since the calculation of these derivatives is easier than for the derivatives of the SIC energy, we expect that an optimization of the Fermi orbital descriptors via the Löwdin eigenvalues will be faster than the present one. Thus, this optimization would accelerate the FLO SIC method as well. If the SIC calculations with the NRLMOL computer code were faster, the method could be applied to larger molecules. Moreover, the FLO SIC method, that is currently only used for molecules and clusters, could then be extended to solids.

A. Appendix This appendix outlines the possible fields of application of second order derivatives in numerical minimization algorithms in Sec. A.1. Moreover, Sec. A.2 reviews the main results of this document.

A.1. Improved Minimization using the Hessian This section will review the usage of second order derivatives for the acceleration of different numerical minimization processes. Since the second order derivative of any function describes its curvature, it is reasonable, that it is useful for finding the extrema of that function. Today, the numerical minimization of any function 𝑓 is done by nonlinear optimization methods. The two mainly used methods are the Limited memory Broyden-Fletcher–GoldfarbShanno (L-BFGS) method and the conjugate gradient (CG) method using a variable metric preconditioner [9, p. 507]. Both methods are based on the same idea. In each iteration step, one first constructs a descent search direction 𝒅𝑘 . Next, a line search is done, that yields the step lengths 𝛼𝑘 . The next iteration point 𝒙𝑘+1 is then given by [10, Eq. 8.3] 𝒙𝑘+1 = 𝒙𝑘 + 𝛼 𝒅𝑘 .

(156)

The CG search direction 𝒅𝑘+1 is constructed from the gradient 𝒈𝑘+1 , i.e. 𝒈𝑘+1 = ∇𝑓(𝒙𝑘+1 ) ,

(157)

the update parameter 𝛽𝑘 and the previous search direction 𝒅𝑘 as 𝒅𝑘+1 = −𝒈𝑘+1 + 𝛽𝑘 𝒅𝑘 .

(158)

Hence, the initial search direction 𝒅0 reads 𝒅0 = −𝒈0 ,


(159)

32

A. Appendix

i.e. the algorithm starts with a search direction that is the direction of steepest descent. There are various update parameters. The current implementation of the NRLMOL computer code uses the Polak-Ribière update parameter 𝛽 PRP , that is given by [3] 𝛽𝑘PRP =

⊺ 𝒈𝑘+1 𝒚𝑘

|𝒈𝑘 |2

,

(160)

where 𝒚𝑘 is the difference of the related gradients of 𝑓, that is 𝒚𝑘 = ∇𝑓(𝒙𝑘+1 ) − ∇𝑓(𝒙𝑘 ) .

(161)

The iteration point 𝒙𝑘 is updated as described, until a certain stop criterion, as e.g. a sufficiently small gradient, is fulfilled. This CG minimization is improved by the usage of a so called preconditioner 𝑃𝑘 . The application of such a preconditioner corresponds to a coordinate transformation into a coordinate system, where the minimization is easier. Such a preconditioner 𝑃𝑘 is a positive definite matrix, that approximates the inverse of the Hessian, i.e. 𝑃𝑘 = 𝐻𝑘−1 = 𝐻 −1 (𝒙𝑘 ) . (162) The Hessian 𝐻 of the function 𝑓 is given by 𝜕 2 𝑓(𝒙) ∣ 2 ∣ 𝜕𝑥1 ∣ ∣ 𝜕 2 𝑓(𝒙) ∣ 𝜕𝑥 𝜕𝑥 𝐻(𝒙) = 𝑓 ″ (𝒙) = ∣ 2 1 ∣ ⋮ ∣ ∣ 2 ∣ 𝜕 𝑓(𝒙) ∣ 𝜕𝑥𝑁 𝜕𝑥1

𝜕 2 𝑓(𝒙) 𝜕𝑥1 𝜕𝑥2

⋯

𝜕 2 𝑓(𝒙) 𝜕𝑥22

⋯

⋮

⋱

𝜕 2 𝑓(𝒙) 𝜕𝑥𝑁 𝜕𝑥2

⋯

𝜕 2 𝑓(𝒙) ∣ 𝜕 2 𝑥1 𝜕𝑥𝑁 ∣ ∣ 𝜕 2 𝑓(𝒙) ∣ 𝜕𝑥2 𝜕𝑥𝑁 ∣ ∣ . ∣ ⋮ ∣ ∣ 𝜕 2 𝑓(𝒙) ∣ 𝜕𝑥2𝑁 ∣

(163)

With that, the search direction 𝒅𝑘+1 is then constructed as (cp. [10, Eq. 5.12]) 𝒅𝑘+1 = 𝑃𝑘 𝒈𝑘+1 + 𝛽𝑘 𝒅𝑘 .

(164)

In case of preconditioning, the update parameter 𝛽𝑘PRP is given by [3] 𝛽𝑘PRP =

⊺ 𝒈𝑘+1 𝑃 𝑘 𝒚𝑘 . ⊺ 𝒈𝑘 𝑃𝑘 𝒈𝑘

(165)

Preconditioned CG is a suitable method for minimization. Usually, the Hessian has to be approximated, e.g. with Broyden-Fletcher–Goldfarb-Shanno (BFGS) (see paragraph below). Once we have obtained analytical expressions for the exact Hessian of the SIC energy, its minimization can be done by an optimized CG, with the best preconditioner one can think of. However, as already mentioned, the preconditioner has to be positive definite. This property cannot be assumed for the Hessian of the SIC energy and hence has to be ensured. We suggest, to transform the matrix into its eigenbasis, where it is diagonal. There, its positive definiteness is guaranteed by setting all eigenvalues to their absolute value, i.e. |𝜆 | 0 𝜆 0 ⋯ 0 ∣ ∣ 1 ∣ 1 0 |𝜆2 | 0 𝜆 ⋯ 0 2 ̃ ∣ ↦ 𝐻(𝒙) =∣ 𝐻(𝒙) = ∣ ⋮ ⋮ ⋱ ⋮ ∣ ∣ ⋮ ∣ ⋮ 0 0 ⋯ 𝜆𝑁 ∣ ∣ 0 ∣0

⋯ 0 ∣ ⋯ 0 ∣ . ⋱ ⋮ ∣ ⋯ |𝜆𝑁 |∣

(166)


A.2. Central Results

33

Next, this matrix 𝐻̃ is transformed back into the original basis. With the inverse of this new matrix 𝐻,̃ a good preconditioner for the CG is given. Nonetheless, there is a second, widely used method for minimization, the L-BFGS method. We will now outline this method and its improvement by analytical expressions of the Hessian. To understand L-BFGS, we start by some introduction to the BFGS algorithm. The BFGS method is the most popular quasi-Newton algorithm [10]. As CG, BFGS creates a search direction 𝒅𝑘 in each iteration step. Now, this is given by [10, Eq. 8.2] −1 𝒅𝑘 = −𝐵𝑘 𝒈𝑘 ,

(167)

where 𝐵𝑘 is a positive definite matrix that approximates the Hessian 𝐻 of 𝑓 at the iteration point 𝒙𝑘 (cp. Eq. (163)) The new iterate 𝒙𝑘+1 is constructed from the former one 𝒙𝑘 by means of Eq. (156), i.e. BFGS minimizes 𝑓 along the search direction 𝒅𝑘 . The matrix 𝐵𝑘 has to be updated at every iteration, since the iteration point is updated, i.e. 𝒙𝑘 ↦ 𝒙𝑘+1 .

(168)

The great power of BFGS is, that it provides the opportunity to calculate 𝐵𝑘 from the former steps. Explicitly, this reads (see [3]) 𝐵𝑘+1 = (1 −

⊺ 𝒚𝑘 𝒔⊺ 𝒔𝑘 𝒔⊺ 𝒔𝑘 𝒚𝑘 𝑘 𝑘 ) 𝐵 ( 1 − ) + . 𝑘 ⊺ ⊺ 𝒔⊺ 𝒚 𝒔 𝒚 𝒔 𝒚 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘

(169)

Here, 𝒔𝑘 denotes the difference of two following iteration steps, i.e. [10, Eq. 8.5] 𝒔𝑘 = 𝒙𝑘+1 − 𝒙𝑘 ,

(170)

whereas 𝒚𝑘 is the same as in CG, given in Eq. (161). As already mentioned, a great advantage is the possibility to calculate 𝐵𝑘 from the former ones, since for most functions that have to be minimized, it is not even possible to find analytical expressions for the Hessian. But Eq. (169) does not give any information how to calculate the first matrix 𝐵0 . For this reason, a initial guess for 𝐵𝑘 is required. A commonly used, simple initial 𝐵𝑘 is the unitary matrix. By contrast, the best possibility to chose is for logical reasons the correct Hessian 𝐻(𝒙0 ), that is the best approximation for itself, i.e. it is the best 𝐵0 . With the known expressions for the Hessian are known, the problem of a starting value for 𝐵𝑘 in BFGS algorithms can be solved. This solution becomes even more important, when we progress from BFGS to L-BFGS. L-BFGS algorithms restart the BFGS iteration after a certain amount of steps. Therefore, the problem of finding some initial Hessian emerges more often then for BFGS and is of much higher importance. All in all, this section showed, that the analytical expressions for the Hessian of the SIC energy do have practical relevance and could help to improve FLOSIC.

A.2. Central Results We will now summarize the main results, i.e. we will give their expressions. The first order derivative of the SIC energy with respect to the Fermi orbital descriptors (see Sec. 3.1) is


34

A. Appendix

given by 𝑁 𝑒2 𝑛 (𝒙′ ) 𝜕 𝑛𝑖 (𝒙) 𝜕 𝐸 SIC =−∑ ∫ d3 𝒙∫ d3 𝒙′ 𝑖 𝜕 𝒂𝑚 4𝜋𝜀0 |𝒙 − 𝒙′ | 𝜕 𝒂𝑚 𝑖=1 𝑁

𝑒2 3 1/3 𝜕 𝑛 (𝒙) 1/3 ∫ d3 𝒙 𝑛𝑖 (𝒙) 𝑖 ( ) 4𝜋𝜀0 𝜋 𝜕 𝒂𝑚 𝑖=1

−∑ 𝑁

= ∑ (⟨ 𝑘=1

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝑉 ̂ SIC ∣ 𝜙𝑘 ⟩ + ⟨𝜙𝑘 ∣ 𝑉𝑘̂ SIC ∣ ⟩) . 𝜕 𝒂𝑚 𝑘 𝜕 𝒂𝑚

(95 revisited) (100 revisited)

The second order derivative reads (see Secs. 3.2 and 3.3) 𝜕 2 𝑛𝑖 (𝒙) 𝜕 2 𝐸 SIC = ∫ d3 𝒙 𝑣𝑖SIC (𝒙) 𝜕 𝒂𝑚 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 2 𝑒 𝜕 𝑛𝑖 (𝒙′ ) 1 𝜕 𝑛𝑖 (𝒙) − ∫ d3 𝒙∫ d3 𝒙′ ′ 4𝜋𝜀0 𝜕 𝒂𝑛 |𝒙 − 𝒙 | 𝜕 𝒂𝑚 −

3 1/3 𝜕 𝑛𝑖 (𝒙) 𝜕 𝑛𝑖 (𝒙) −2/3 1 𝑒2 ( ) ∫ d3 𝒙 𝑛 (𝒙) . 3 4𝜋𝜀0 𝜋 𝜕 𝒂𝑛 𝜕 𝒂𝑚 𝑖

(135 revisited)

This can also be written as 𝜕 2 𝐸 SIC = 𝐴(𝒂𝑛 , 𝒂𝑚 ) + 𝐵(𝒂𝑛 , 𝒂𝑚 ) , 𝜕 𝒂𝑚 𝜕 𝒂 𝑛

(144 revisited)

where the first term 𝐴 reads 𝑁

𝐴(𝒂𝑛 , 𝒂𝑚 ) = 2 ∑ 𝜆𝑘 𝑗𝑘 ⟨ 𝑗,𝑘=1

𝜕 2 𝜙𝑘 ∣𝜙 ⟩ 𝜕 𝒂𝑚 𝜕 𝒂 𝑛 𝑗

𝑁

+ 2 ∑ 𝜆𝑘 𝑗𝑙 ⟨ 𝑗,𝑘,𝑙=1

𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑙 ∣ ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑛

(154 revisited)

with the definitions ̂ SIC ∣ 𝜙𝑙 ⟩ , 𝜆𝑘 𝑗𝑙 ∶= ⟨𝜙𝑗 ∣ 𝑉𝑘

(152 revisited)

and ⟨

3 𝜕 𝜙𝑘 𝜕 𝜙𝑘 ∣ 𝜙𝑗 ⟩ ⟨𝜙𝑙 ∣ ⟩ = ∑ ⟨𝐷ℎ,𝑘𝑚 | 𝜙𝑗 ⟩ ⟨𝜙𝑙 | 𝐷𝑖,𝑘𝑛 ⟩ . 𝜕 𝒂𝑚 𝜕 𝒂𝑛 ℎ,𝑖=1

(155 revisited)

The quantities ⟨𝜙𝑙 | 𝐷1,𝑘𝑚 ⟩, ⟨𝜙𝑙 | 𝐷2,𝑘𝑚 ⟩ and ⟨𝜙𝑙 | 𝐷3,𝑘𝑚 ⟩ are defined in Eqs. (116), (118) and (128). The derivative of the Löwdin eigenvalues with respect to the Fermi orbital descriptors reads 𝑁 𝑁 𝑁 𝜕 𝑆𝑖𝑗 𝜕 𝑇𝛼𝑗 𝜕 𝑄𝛼 𝜕 𝑇𝛼𝑖 = ∑ 𝑆𝑖𝑗 𝑇𝛼𝑗 + ∑ 𝑇𝛼𝑖 𝑇𝛼𝑗 + ∑ 𝑇𝛼𝑖 𝑆𝑖𝑗 . 𝜕 𝒂𝑚 𝑖,𝑗=1 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝜕 𝒂𝑚 𝑖,𝑗=1 𝑖,𝑗=1

(122 revisited)

Furthermore, 𝐵 is given by 𝑁

𝐵(𝒂𝑛 , 𝒂𝑚 ) = ∑ (⟨ 𝑘=1

𝜕 𝑉𝑘̂ SIC 𝜕 𝜙𝑘 𝜕 𝜙𝑘 𝜕 𝑉𝑘̂ SIC ∣ ∣ 𝜙𝑘 ⟩ + ⟨𝜙𝑘 ∣ ∣ ⟩) . 𝜕 𝒂 𝑚 𝜕 𝒂𝑛 𝜕 𝒂 𝑛 𝜕 𝒂𝑚

(147 revisited)

These formulae are the main result of this work. : Analytical second order derivatives in Fermi Löwdin orbital self-interaction correction

References

35

References [1] Torsten Fliessbach. Elektrodynamik. Lehrbuch zur Theoretischen Physik 2. Spektrum Akademischer Verlag, 2005. [2] Gabriele Giuliani and Giovanni Vignale. Quantum Theory of the Electron Liquid. Cambridge University Press, 2005. [3] William W. Hager and Hongchao Zhang. A survey of nonlinear conjugate gradient methods. Pacific journal of Optimization, 2(1), 2006. [4] Torsten Hahn, Simon Liebing, Jens Kortus, and Mark R. Pederson. Benchmarking fermi orbital self-interaction corrected density functional theory on molecules. arXiv preprint arXiv:1508.00745, 2015. [5] Torsten Hahn, Simon Liebing, Jens Kortus, and Mark R. Pederson. Fermi orbital selfinteraction corrected electronic structure of molecules beyond local density approximation. The Journal of Chemical Physics, 143(22), 2015. [6] Jens Kortus and Mark R. Pederson. Magnetic and vibrational properties of the uniaxial Fe13 O8 cluster. Phys. Rev. B, 62(9), 2000. [7] Philippe A. Martin and François Rothen. Many-body Problems and Quantum Field Theory: An Introduction. Springer, 2002. [8] Richard M. Martin. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004. [9] John L. Nazareth. Conjugate gradient methods less dependent on conjugacy. SIAM review, 28(4), 1986. [10] Jorge Nocedal and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006. [11] Mark R. Pederson. Fermi orbital derivatives in self-interaction corrected density functional theory: Applications to closed shell atoms. The Journal of Chemical Physics, 142(6), 2015. [12] Mark R. Pederson and Tunna Baruah. Chapter Eight - Self-Interaction Corrections Within the Fermi-Orbital-Based Formalism . volume 64 of Advances In Atomic, Molecular, and Optical Physics, pages 153 – 180. Academic Press, 2015. [13] Mark R. Pederson, Tunna Baruah, Der-You Kao, and Luis Basurto. Self-interaction corrections applied to Mg-porphyrin, C60, and pentacene molecules. The Journal of Chemical Physics, 144(16), 2016. [14] Mark R. Pederson, Richard A. Heaton, and Joseph G. Harrison. Metallic state of the free-electron gas within the self-interaction-corrected local-spin-density approximation. Phys. Rev. B, 39(3), Jan 1989. [15] Mark R. Pederson, Richard A. Heaton, and Chun C. Lin. Local‐density Hartree–Fock theory of electronic states of molecules with self‐interaction correction. The Journal of Chemical Physics, 80(5), 1984.


36

References

[16] Mark R. Pederson, Richard A. Heaton, and Chun C. Lin. Density‐functional theory with self‐interaction correction: Application to the lithium molecule. The Journal of Chemical Physics, 82(6), 1985. [17] Mark R. Pederson and Koblar A. Jackson. Variational mesh for quantum-mechanical simulations. Physical Review B, 41(11), 1990. [18] Mark R. Pederson and Koblar A. Jackson. Pseudoenergies for simulations on metallic systems. Physical Review B, 43(9), 1991. [19] Mark R. Pederson and Chun C. Lin. Localized and canonical atomic orbitals in self‐interaction corrected local density functional approximation. The Journal of Chemical Physics, 88(3), 1988. [20] Mark R. Pederson, Dirk V. Porezag, Jens Kortus, and David C. Patton. Strategies for massively parallel local-orbital-based electronic structure methods. physica status solidi (b), 217(1), 2000. [21] Mark R. Pederson, Adrienn Ruzsinszky, and John P. Perdew. Communication: Selfinteraction correction with unitary invariance in density functional theory. The Journal of Chemical Physics, 140(12), 2014. [22] John P. Perdew. Density functional theory and the band gap problem. International Journal of Quantum Chemistry, 28(S19), 1985. [23] John P. Perdew, John A Chevary, Sy H. Vosko, Koblar A. Jackson, Mark R. Pederson, Dig J. Singh, and Carlos Fiolhais. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Physical Review B, 46(11), 1992. [24] John P. Perdew and Alex. Zunger. Self-interaction correction to density-functional approximations for many-electron systems. Physical Review B, 23(10), 1981. [25] Dirk Porezag. Development of Ab-Initio and Approximate Density Functional Methods and their Application to Complex Fullerene Systems. PhD thesis, TU Chemnitz, Fakultät für Naturwissenschaften, 1997. [26] Dirk Porezag and Mark R. Pederson. Optimization of Gaussian basis sets for densityfunctional calculations. Physical Review A, 60(4), October 1999. [27] Adrienn Ruzsinszky, John P. Perdew, Gábor I. Csonka, Oleg A. Vydrov, and Gustavo E. Scuseria. Spurious fractional charge on dissociated atoms: Pervasive and resilient selfinteraction error of common density functionals. The Journal of Chemical Physics, 125(19), 2006. [28] Ronald Starke and Giulio A. H. Schober. Response Theory of the Electron-Phonon Coupling. ArXiv e-prints, May 2016.


Self-interaction correction to electronic structure ...

Self-interaction correction to electronic structure ...

Suggest Documents

Introduction to electronic structure simulations

Introduction to Electronic Structure Theory

Introduction to electronic structure simulations

Correction of Structure Determinations on

Imaging from atomic structure to electronic structure

Correction to: electronic cigarette use behaviors ... - BMC Public Health

The Electronic Structure ot

Electronic structure and photoluminescence

Electronic structure of CaFe2As2

REAL-SPACE ELECTRONIC STRUCTURE

Electronic structure of FeSi

Crystal structure, electronic structure, temperature ...

Crystal structure, electronic structure, optical and ...

Capital Structure Model (CSM): correction, constraints

Electronic-structure-based molecular-dynamics

Electronic structure of overstretched DNA

Electronic Structure of Carbon NanoparticlesÂ¶

XAS Results on Electronic Structure

Electronic Structure of Crystalline Materials

Bi3TeBO9: electronic structure, optical properties

Electronic structure of ThBe 13

Electronic structure of Sn2P2S6 - JuSER

Electronic Structure of Carbon Nanocones

ELECTRONIC STRUCTURE AND PARTIAL CHARGE