Viraht Sahni Quantal Density Functional Theory Second Edition Quantal Density Functional Theory Viraht Sahni Quantal Density Functional Theory Second Edition 123 Viraht Sahni Brooklyn, NY USA ISBN 978-3-662-49840-8 DOI 10.1007/978-3-662-49842-2 ISBN 978-3-662-49842-2 (eBook) Library of Congress Control Number: 2016939990 © Springer-Verlag Berlin Heidelberg 2003, 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg ’Tis not nobler in the mind to suffer The slings and arrows of outrageous fortune, ’Tis nobler, and ennobling, To get off the ground and fight like hell In Memoriam My beloved mother and father, Hema and Harbans Lal Preface to the Second Edition The idea of writing a second edition within slightly more than a decade of the publication of the first is a consequence of the considerable new understandings of Quantal Density Functional Theory (Q–DFT) achieved over this period But there have also been further insights into Schrödinger theory, and to the significance of the first theorems of Hohenberg-Kohn and Runge-Gross density functional theory (DFT) The book is still comprised of the three principal components: a description of Schrödinger theory from the new perspective of the ‘Quantal Newtonian’ second and first laws for the individual electron; traditional Hohenberg-Kohn, Runge-Gross, and Kohn-Sham density functional theory; and Q– DFT together with applications to explicate the theory, and the physical insights it provides into traditional DFT, Slater theory, and local effective potential energy theory in general However, each component has been revised to incorporate the new understandings Then there is the new material on the extension of Q–DFT to the added presence of an external magnetostatic field It was the attempt to extend the theory to the presence of magnetic fields that forced the reexamination of both traditional DFT and Q–DFT, thereby leading to many of the new insights The extension to external magnetic fields required a critical reevaluation of the existing literature This in turn led to the proof of the corresponding Hohenberg-Kohn theorems for uniform magnetostatic fields, one that is distinct from but in the rigorous sense of the original The Q–DFT in a magnetic field is then explicated by an example in two-dimensional space Working on the second edition has been akin to writing a new book The pedagogical nature of the book has been maintained Most of the new derivations are once again given in detail And as a result of the new understandings, it has been possible to present Q–DFT for arbitrary external electromagnetic fields whether they be time-dependent or time-independent in a most general and comprehensive manner The common thread of the ‘Quantal Newtonian’ laws for the individual electron is now weaved throughout the book Xioayin Pan has been a principal contributor to the new developments Our collaboration has been productive, and working with Xiaoyin has been a pleasure ix x Preface to the Second Edition Together with Doug Achan, a former graduate student, and Lou Massa, a friend and colleague, new physics of the Wigner low-density high-electron correlation regime of a nonuniform density system has been discovered Thus, an additional characterization of the Wigner regime is proposed The example studied also provides a contrast to the high-density low-electron correlation regime of atoms and molecules Thanks are also due to Xiaoyin and Lou for their critical comments on various chapters Once again I wish to acknowledge Brooklyn College for the support and freedom afforded to me to pursue the research of my interest Finally, with much gratitude, I wish to thank my wife Catherine for typing the book despite the travails of life Brooklyn, NY, USA Viraht Sahni Preface to the First Edition The idea underlying this book is to introduce the reader to a new local effective potential energy theory of electronic structure that I refer to as Quantal Density Functional Theory (Q–DFT) It is addressed to graduate students who have had a one year course on Quantum Mechanics, and to researchers in the field of electronic structure It is pedagogical, with detailed proofs, and many figures to explain the physics The theory is based on the first Hohenberg–Kohn theorem, and is distinct from Kohn–Sham density functional theory No prior understanding of traditional density functional theory is required as the theorems of Hohenberg and Kohn, and Kohn–Sham theory, and their extension to time-dependent phenomenon are described There are other excellent texts on traditional density functional theory, and as such I have kept the overlap with the material in these texts to a minimum It is also possible via Q–DFT to provide a rigorous physical interpretation of Kohn– Sham theory and other local effective potential energy theories such as Slater theory and the Optimized Potential Method A second component to the book is therefore the description and the explanation of the physics of these theories My interest in density functional theory began in the early 1970s simultaneously with my work on metal surface physics The origins of Q–DFT thus lie in my attempts to understand the physics underlying the formal framework of Kohn– Sham density functional theory and of various approximations within it in the context of the nonuniform electron gas at a metal surface My work with Manoj Harbola [1, 2] constitutes the ideas seminal to Q–DFT The history of how these ideas developed, and of their evolution to Q–DFT, is a classic example of how science works This is not the place to describe the many twists and turns in the path to the final version of the theory However, together with a further understanding [3] noted, credit must also be afforded Andrew Holas and Norman March whose work [4] helped congeal and close the circle of ideas I wish to gratefully acknowledge my graduate students Cheng Quinn Ma, Abdel Mohammed, Manoj Harbola, Marlina Slamet, Alexander Solomatin, Zhixin Qian, and Xiaoyin Pan whose creative work has contributed both directly and indirectly to the writing of this book xi xii Preface to the First Edition Then there is my friend and colleague Lou Massa whose enthusiasm for the subject matter of the book and whose consistent support and critique during its writing have proved invaluable Brooklyn College has been home, and I thank the College for its support of my research The book was typed by Suzanne Whiter, throughout with a smile To her my heartfelt thanks To my wife, Catherine, I owe an immense debt of gratitude She has suffered happily over the years through the many referee reports of my papers I thank her for being there with me every step of the way Brooklyn, NY, USA October 2003 References M.K Harbola, V Sahni, Phys Rev Lett 62, 489 (1989) V Sahni, M.K Harbola, Int J Quantum Chem 24, 569 (1990) V Sahni, M Slamet, Phys Rev B 48, 1910 (1993) A Holas, N.H March, Phys Rev A 51, 2040 (1995) Viraht Sahni 398 Appendix H: Derivation of the Kinetic-Energy-Density Tensor … and ∂ − (r 2p +r22 ) 2 e = −r pα e− (r p +r2 ) , ∂r pα (H.6) the resulting tαβ [r; γ ] is a sum of terms listed below: Term = C e−r Term = C e−r rβ Term = C e−r rα (r2α − rα )(r2β − rβ ) −r22 e dr2 , |r2 − r|2 (r2α − rα ) (1 + |r2 − r|)e−r2 dr2 , |r2 − r| (r2β − rβ ) (1 + |r2 − r|)e−r2 dr2 , |r2 − r| Term = C e−r rα rβ (1 + |r2 − r|)2 e−r2 dr2 , 2 = rα rβ ρ(r ), (H.7) (H.8) (H.9) (H.10) (H.11) where ρ(r ) is the electron density given in (G.1) Next in (H.7) transform the coordinates to r3 = r2 − r Then Term = C e−r r3α r3β −(r +r32 +2r·r3 ) e dr3 r32 (H.12) Since r3α r3β e−2r·r3 = ∂2 e−2r·r3 , ∂rα ∂rβ (H.13) then Term = 2π −2r ∂ C e ∂rα ∂rβ ∞ −r32 e I0 (2rr3 )dr3 , r3 (H.14) where 2π e−2rr3 cos θ3 dθ3 = 2π I0 (2rr3 ), (H.15) and I0 (r ) the zeroth-order Bessel function To eliminate the singularity at r3 = 0, employ 2rβ r3 ∂ I1 (2rr3 ), I0 (2rr3 ) = ∂rβ r (H.16) Appendix H: Derivation of the Kinetic-Energy-Density Tensor … 399 where I1 (r ) is the first-order Bessel function Thus, we have ∂ rβ ∞ −r32 e I1 (2rr3 )dr3 ∂rα r ∂ = πC e−2r rβ f (r ) , ∂rα Term = πC e−2r (H.17) (H.18) where f (r ) = er − 2r (H.19) Now for a general function f (r ), ∂ rα rβ ∂ f (r ) [rβ f (r )] = δαβ f (r ) + ∂rα r ∂r (H.20) Thus, (H.18) is Term = δαβ [πC e−2r f (r )] + rα rβ ∂ f (r ) πC e−2r r ∂r (H.21) The steps to obtain Terms and 3, which are identical, are the same as described above: apply the coordinate transformation, employ r3β exp(−2r · r3 ) = −( 21 )∂ [exp −2r·r3 ]/∂rβ , and rβ ∂ f (r )/∂rα = (rα rβ /r )∂ f (r )/∂r where f (r ) is any function of r , to obtain Term (2 + 3) = − rα rβ r 2πC e−2r ∂ f (r ) , ∂r (H.22) where √ π r2 r r2 e I0 ( ) f (r ) = e + 2 On summing (H.11), (H.21) and (H.22), one obtains (G.14) for tαβ (r; γ ) References T Yang, X.-Y Pan, V Sahni, Phys Rev A 83, 042518 (2011) M Slamet (private communication) (H.23) Appendix I Derivation of the Pair–Correlation Density in the Local Density Approximation for Exchange In this appendix we derive [1] the analytical expression for the pair–correlation density gxL D A {rr ; ρ(r)} in the local density approximation (LDA) for exchange by the method of Kirzhnits [2] For a Slater determinant {φi } of orbitals φi (r), the pair–correlation density gx (rr ) = ρ(r ) + ρx (rr ) Both the density ρ(r) and the Fermi hole charge ρx (rr ) are defined in terms of the idempotent Dirac density matrix γs (rr ), which with the spin index suppressed is φ ∗j (r)φ j (r ) (I.1) ρx (rr ) = −|γs (rr )|2 /2ρ(r) (I.2) γs (rr ) = j: j ≤ F Thus, ρ(r) = γs (rr), and The orbitals φ j (r) are in turn solutions of the S system type differential equation: − ∇ + vs (r) φi (r) = i φi (r), (I.3) with vs (r) a local multiplicative operator Therefore, to obtain gxL D A {rr ; ρ(r)}, one must expand the density matrix γs (rr ) in gradients of the density about the uniform electron gas result The density matrix is first written in terms of the Fermi energy F as ∞ γs (rr ) = ∗ j )φ j (r)φ j (r ( F − ) ( F − tˆ − vˆs )φ ∗j (r)φ j (r ), j=1 ∞ = (I.4) j=1 © Springer-Verlag Berlin Heidelberg 2016 V Sahni, Quantal Density Functional Theory, DOI 10.1007/978-3-662-49842-2 401 402 Appendix I: Derivation of the Pair–Correlation Density in the Local Density … where (x) is the step function, and tˆ = − 21 ∇ the kinetic energy operator Defining the operator TˆF (r) = 21 k 2F (r) for the local Fermi energy TF (r) = F − vs (r), the density matrix can then be written as γs (rr ) = (TˆF − tˆ) ∞ φ ∗j (r)φ j (r ) (I.5) j=1 ∗ Employing the completeness of the single–particle orbitals φ j (r) i.e ∞ j=1 φ j (r)φ j (r ) = δ(r − r ) together with the representation of the delta function δ(r − r ) in terms of plane–wave orbitals one obtains γs (rr ) = (2π )3 dk (TˆF − tˆ)eik·r e−ik·r , (I.6) where the factor of is for the spin The step function (TˆF − tˆ) ≡ (kˆ 2F − kˆ ) can ˆ where f = , aˆ = −kˆ = ∇ , and bˆ = k 2F (r) Thus simply be written as f (aˆ + b), (I.6) becomes γs (rr ) = (2π )3 ˆ ik·r e−ik·r dk f (aˆ + b)e (I.7) This leads to a mathematical problem of the following general nature: given the eigenfunction |a of an operator a, ˆ where a|a ˆ = a|a with |a = eik·r and a = −k , ˆ how can one compute the quantity f (aˆ + b)|a if the operator aˆ does not commute ˆ = To tackle this problem, f (aˆ + b)|a ˆ with bˆ : [a, ˆ b] is rewritten in terms of its Laplace (or Fourier) transformation as ˆ f (aˆ + b)|a = ˆ )|a , dτ F(τ ) E(τ (I.8) ˆ ˆ b) ˆ ) = eτ (a+ Since aˆ does not where τ is a real (or imaginary) parameter, and E(τ ˆ the operator E(τ ˆ ) does not simply equal to (eτ aˆ eτ bˆ ) nor (eτ bˆ eτ aˆ ) commute with b, ˆ ) can be put into Thus, a supplementary operator Kˆ is introduced such that E(τ normal form, which means a product in which all kˆ operators are to the right of the rˆ operators, so that the kˆ and rˆ operators can be treated as classical variables: ˆ ˆ ˆ b) ˆ ) = eτ (a+ = eτ b Kˆ (τ )eτ aˆ E(τ (I.9) Thus ˆ )|a = eτ bˆ Kˆ (τ )eτ aˆ |a = eτ bˆ Kˆ (τ )eτ a |a , E(τ (I.10) Appendix I: Derivation of the Pair–Correlation Density in the Local Density … 403 and consequently ˆ ˆ ˆ )|a = eτ (a+b) K (τ )|a E(τ (I.11) Note that the eigenvalue a has now replaced the operator aˆ in the argument of the exponential function, and consequently it now commutes with the operator bˆ such ˆ appears only in the operator Kˆ (τ ) that the dependence on the commutators [a, ˆ b] Now in order to determine Kˆ (τ ), one must first determine an expression for the differential equation of Kˆ (τ ) by differentiating both sides of (I.9) with respect to τ One then obtains ∂ Kˆ ˆ ˆ ˆ τ b Kˆ − Kˆ a ˆ = e−τ b ae ∂τ (I.12) The expansion of Kˆ in powers of τ is obtained by expanding both of the exponential functions of (I.12) in a Taylor series This results in the expansion ˆ ˆ ˆ2 τ b − τ bˆ + · · · 2! 3! 1 aˆ + τ bˆ + τ bˆ + τ bˆ + · · · 2! 3! ˆ b, ˆ aˆ − · · · ˆ aˆ + τ b, = aˆ − τ b, 2! e−τ b ae ˆ τ b = − τ bˆ + (I.13) In general form ˆ ˆ e−τ b ae ˆ τb = ∞ n=0 (−τ )n ˆ ˆ ˆ ˆ aˆ b, b, b, , b, n! n times (I.14) − ··· Kˆ − Kˆ a ˆ (I.15) By substituting (I.13) into (I.12) one obtains ∂ Kˆ ˆ b, ˆ aˆ + τ b, ˆ aˆ = aˆ − τ b, ∂τ 2! The solution Kˆ (τ ) of this differential equation is determined via iteration whereby the (i +1)th order solution of Kˆ is obtained from the ith order solution by substituting the latter into the right hand side of (I.15), and then integrating the differential equation Thus Kˆ i+1 = (∂ Kˆ i+1 /∂τ )dτ = Oˆ K i dτ , where Oˆ Kˆ i is the entire right hand side of (I.15) with Kˆ i substituted in it The zeroth order solution of Kˆ implies that there ˆ a], is no dependence on the commutator [b, ˆ and thus it is determined by substituting τ = into (I.9), which results in: Kˆ = The first order solution of Kˆ which implies ˆ a] that only the commutator [b, ˆ is considered to be non-zero, is then obtained as the integral over dτ of (I.12) which is then 404 Appendix I: Derivation of the Pair–Correlation Density in the Local Density … ∂ Kˆ ˆ aˆ = aˆ − τ b, ∂τ ˆ aˆ , = −τ b, Kˆ − Kˆ aˆ (I.16) and which results in ˆ aˆ Kˆ (τ ) = − τ b, (I.17) Then by using (I.8), (I.11), and (I.17) one obtains f aˆ + bˆ |a = dτ F(τ )e − τ a+bˆ dτ F(τ )τ e |a τ a+bˆ ˆ aˆ |a b, (I.18) Since the parameter τ acts as an operator for differentiating the function f with respect to its argument, the right hand side of (I.18) can be rewritten as f aˆ + bˆ |a = f a + bˆ |a − f a + bˆ ˆ aˆ |a , b, (I.19) where f a + bˆ = and k 2F − k , f a + bˆ = δ k 2F − k , f a + bˆ = δ k 2F − k (I.20) ˆ a] The commutator [b, ˆ acting on |a in the second term of the right hand side of ˆ (I.19), i.e [b, a]|a ˆ = −[k 2F , kˆ ]eik·r , is evaluated by employing the relationship kˆ = 1i ∇ so that ˆ aˆ |a = − ∇ k 2F + 2i∇k 2F · k eik·r b, (I.21) Since we are interested in determining the density matrix γs (rr ) only to lowest order in ∇, we drop the first term in the parentheses Then by using (I.7), (I.19)–(I.21) we obtain γs (rr ) = (2π )3 dk (k 2F − k ) + δ (k 2F − k )2i∇k 2F · k eik·(r−[r ) (I.22) Appendix I: Derivation of the Pair–Correlation Density in the Local Density … 405 The integral of this equation can easily be solved by shifting the origin of the position vector r to the tip of the position vector r, such that R = r − r, and then choosing the direction of R along the z-axis such that k · (r − r ) = −k R cos θ The first term of the integral of (I.22) is evaluated in a straight forward manner, and results in (2π )3 dk (k 2F − k )eik·(r−r ) = k 3F j1 (k F R) , π 2k F R (I.23) where j1 (x) = sin x − x cos x x2 (I.24) is the first–order spherical Bessel function The second term of the integral of (I.22) is evaluated by partial integration and by rewriting δ (k 2F − k ) = − ∂ δ(k 2F − k ) 2k ∂k (I.25) in order to first eliminate the first derivative of the delta function Then by employing the relation δ(x − xi ) , | f (xi )| (I.26) δ(k − k F ) δ(k + k F ) + 2k F 2k F (I.27) δ[ f (x)] = i where f (xi ) = 0, f (xi ) = 0, we have δ(k 2F − k ) = The second delta function on the right does not contribute to the integral so that finally one obtains (2π )3 dk δ (k 2F − k2 )2i∇k 2F · k eik·(r−r ) = (∇k 2F · R) sin(k F R)/4π (I.28) Thus, to first order in ∇, the Dirac density matrix is γs (rr ) = k 3F j1 (k F R) ˆ sin(k F R), (∇k 2F · R) + π kF R 4π (I.29) ˆ = R/R It is thus evident from (I.29) that the density ρ(r) = γs (rr) is of with R O(∇ ) to lowest order in the gradients of the density Since j1 (x) ∼ x/3 for small x, then to O(∇ ) the density ρ(r) and local Fermi momentum k F (r) are related by 406 Appendix I: Derivation of the Pair–Correlation Density in the Local Density … the first term of (I.29), so that ρ(r) = k 3F (r)/3π which is the uniform electron gas relationship Finally by considering the expansions of γs (rr ) and ρ(r ) to terms of 0(∇) assumed valid locally, one obtains from (I.28), (I.2) and (I.29) the expression for the LDA pair–correlation density as gxL D A {rr ; ρ(r)} = k 3F (r ) + ρxL D A {rr ; ρ(r)} 3π (I.30) where the Fermi hole in the LDA is given by ρxL D A rr ; ρ(r) =− ρ(r) j12 (k F R) j0 (k F R) j1 (k F R) ˆ + ( R · ∇k 2F ) , (k F R)2 2k 3F (I.31) sin x x (I.32) and where j0 (x) = is the zeroth–order spherical Bessel function References V Sahni, in Recent Advances in Density Functional Methods, Part I, ed by D.P Chong (World Scientific, 1995) D.A Kirzhnits, Field Theoretic Methods in Many-Body Systems (Pergamon, London, 1967) Index A Action functionals, 11 Additive constant, 171 Adiabatic coupling constant perturbation theory, 11, 186, 191, 196 Adiabatic coupling constant scheme, 186 Asymptotic structure in the classically forbidden region, 42 of the S system orbital densities, 237 of the density, 44, 237 of the wavefunction, 44 Asymptotic structure with classically forbidden region correlation-kinetic field, 110 correlation-kinetic potential, 111 Coulomb field, 104 Hartree field, 103 Pauli field, 103 Atoms, 210 B Band gap of semiconductors, 231, 232 Band structure of semiconductors, 231 Basic variable, 70, 144, 164, 255, 284, 343 Basic variable of quantum mechanics, Bessel function (spherical) first–order, 318, 405 Bessel function (spherical) zeroth–order, 321, 406 Bohr magneton, 273 Born–Oppenheimer approximation, 16 Bose-Einstein condensates, 256 Brillouin’s theorem, 123 B system, 216 correlation–kinetic energy, 226 correlation–kinetic field, 225 density functional theory of, 218 density matrix, 225 differential equation, 219, 221 discontinuity in the electron–interaction potential, 231 effective field, 225, 227 electron–interaction energy, 219 electron-interaction field, 225 electron-interaction potential, 219, 225 internal field, 226 kinetic energy, 218, 226, 227 kinetic–energy–density tensor, 225 kinetic field, 225 kinetic ‘force’, 225 quantal density functional theory of, 224 wavefunction, 216 C Canonical angular momentum, 259, 264 Canonical momentum, 257 Canonical orbital angular momentum, 7, 254 Chemical potential, 146, 219, 235 ‘Classical’ fields, 7, 15, 22 correlation–current–density field, 78 correlation–kinetic field, 77 Coulomb, 77 current density field, 24 current density for S system, 78 differential density, 57, 78 differential density field, 23 electron–interaction, 76 electron–interaction field, 22, 55 Hartree, 76 Hartree field, 23, 53 kinetic field, 23 kinetic (S system), 77 © Springer-Verlag Berlin Heidelberg 2016 V Sahni, Quantal Density Functional Theory, DOI 10.1007/978-3-662-49842-2 407 408 Pauli, 76 Pauli–Coulomb, 23 Pauli–Coulomb field, 55 Coalescence conditions, 40 Coalescence constraints, 40 cusp coalescence condition, 42 differential form, 40 electron–electron coalescence, 40 electron–nucleus coalescence, 40 electron–nucleus in terms of the density, 41 integral form, 41 node coalescence condition, 42 Conditional probability amplitude, 221 Conduction band, 232 Conservative effective field, Conservative field, 28 Constrained-search, 159 Continuity equation, 31, 60, 63, 262 Correlation–Current–Density effects, 3, 72 Correlation–Current–Density field, 76 Correlation–Kinetic effects, 3, 4, 72, 97 Correlation–Kinetic energy, 95 Correlation–Kinetic field, 76 Correlation-Magnetic effects, Correspondence principle, 256, 259 Coulomb correlations, Coulomb energy, 95 Coulomb field, 75 Coulomb hole, 73, 75 Coulomb repulsion, 16 Coulomb’s law, 22, 76 Coulomb species, 173 Current density, 2, 17, 21, 75 for S system, 78 Current density field, 22 Current density functional theory, 277, 279 Current density in magnetic field diamagnetic component, 261, 277 magnetization component, 277 paramagnetic component, 260, 277 physical, 260 Cyclotron resonance, 256 D Degenerate time-dependent Hamiltonians, 180 Degenerate time-independent Hamiltonians, 171, 178 De Haas van Alphen effect, 256, 284 Density amplitude, 6, 216 Derivative discontinuity, 11, 234 Index Diamagnetic current density, Differential density field, 22 Differential virial theorem, 349 Differential virial theorem for the timeindependent case, 353 Dirac density matrix, 72, 73 gradient expansion, 401 Dirac spinless single-particle density matrix, 73 gradient expansion, 320 idempotency, 73 Discontinuity in the electron–correlation potential correlation contributions due to Quantal density functional theory, 242 Discontinuity in the electron–interaction potential, 232, 247 correlation contributions according to Kohn–Sham theory, 239 correlation contributions due to Quantal density functional theory, 243 expression according to Quantal density functional theory, 243 in terms of S system eigenvalues, 236 in terms of the functional derivative of the energy, 235 origin of the discontinuity, 232 Discontinuity in the electron–interaction potential energy, 231 correlation contributions according to Kohn–Sham theory, 250 correlation contributions according to Quantal density functional theory, 242 Discontinuity of the electron–interaction potential energy in terms of S system eigenvalues, 236 Dissociation of molecules, 231 E Effective field, 83, 97 Effective local electron-interaction potential energy, Ehrenfest’s theorem, 7, 32, 34, 86, 342 for S system, 86 Electron affinity, 234 Electron density, 284 Electronic density, 2, 17, 73, 288 gradient expansion, 321 Electron–interaction energy, 55, 95 Electron-interaction field, 75 Electron–interaction potential energy, 17 Energy components, 25 Index correlation–kinetic, 80 Coulomb, 79 electron–interaction, 25, 79 electron–interaction for S system, 79 external, 80 external potential, 27 Hartree, 79 Hartree or Coulomb self-energy, 25 kinetic, 26 kinetic for S system, 80 Pauli, 79 Pauli–Coulomb, 25 Energy functional of the orbitals, 207 Ensemble density, 233, 234 Ensemble density matrix, 233, 241 Euler equation, 16, 60, 63 Euler–Lagrange equation, 9, 145, 165, 166, 218, 235, 255, 258 External field, 28, 32, 284 External magnetostatic field, External potential energy, 17 External potential energy operator, F Fermi–Coulomb hole, 17, 314 Fermi–Coulomb hole charge, 21, 51 Fermi hole, 73, 315 Fermi hole charge, 74 Fermi momentum, 318 Feynman Path Integral method, 37, 357 Field momentum, 257 Force equation, 31 ‘Forces’ differential density, 23, 57 electron interaction, 22, 76 kinetic, 24, 56 kinetic (S system), 77 Lorentz, 287 Fourier transform, 402 Fractional charge, 231 Fractional number of electrons, 233 Fractional particle number, 233 G Gauge function, Gauge invariance, 153 Gauges Coulomb, 261 Landau, 285 symmetrical, 285 Gauge transformation, 151, 169 Gauge variance, 153 409 Green’s function, 205 Gunnarsson-Lundqvist theorem, 10, 138 Gyromagnetic ratio, 273 H Hall effect, 256, 284 Harmonic Potential Theorem, 36, 45, 179, 355 Hartree energy, 55, 95 Hartree field, 22, 75 Hartree–Fock theory, 5, 118, 325, 344 four theorems, 123 Slater–Bardeen interpretation, 120 Hartree–Fock–Slater differential equation, 332 Hartree theory, 5, 126 Hermitian operator, 18 Hierarchy within the fundamental theorem of density functional theory, 170 High-electron-correlation regime, Highest occupied eigenvalue, Hohenberg–Kohn–Sham density functional theory, 68 Hohenberg-Kohn theory, 2, 140, 343 corollary, 9, 172 first theorem, 2, 141 generalization, 148 generalization to external electrostatic and magnetostatic fields, 253, 265 inverse maps, 154 primacy of electron number, 146 second theorem, 145 Hooke’s atom, 6, 44, 365 first excited singlet state, 44, 46, 99 first excited singlet state properties, 369 ground state, 44, 46, 99 ground state properties, 365 kinetic-energy-density tensor, 375 properties, 56 Hooke’s atom in a magnetic field, 7, 295 correlation-kinetic field and energy, 303 density, 296 electron-interaction field and energy, 301 ground state properties, 391 highest occupied eigenvalue of S system, 310 kinetic-energy-density tensor, 397 physical current density, 296 potentials, 305 single-Particle expectation values, 310 Hooke’s species, 172, 174, 176, 178, 180 410 I Infimum, 156 Integral virial theorem, 29, 86 for S system, 86 time-independent, 29, 39 Internal field, 7, 28, 32, 284 Internal field of the electrons, 57 Intrinsic constant, 171 Ionization energy, 237 Ionization potential, 5, 43, 44, 57, 235 J Jellium metal clusters, 94 Jellium metal surfaces, 94, 208 K Kinetic energy, 17, 95 Kinetic energy density, 26 Kinetic-energy-density tensor, 24, 56, 80 for S system, 77, 80 Kinetic field, 22, 80 Kirzhnits method, 321, 401 Kohn–Sham density functional theory electron–interaction energy functional, 160 electron– interaction potential energy, 161 scaling relationships, 195 within adiabatic coupling constant scheme, 194 Kohn–Sham theory, 9, 158 ‘correlation’, 11 ‘correlation’ energy functional, 162 ‘correlation’ potential, 162 electron-interaction energy functional, ‘exchange’, 11 ‘exchange–correlation’ energy functional, 161 ‘exchange–correlation’ potential, 162 ‘exchange’ energy functional, 162 ‘exchange’ potential, 162 functional derivative, 10 integral virial theorems, 162 interms of adiabatic coupling, 196 ‘exchange correlation’ energy and potential, 186 ‘exchange’ energy and potential, 186 electron-interaction energy and potential, 186 interms of adiabatic coupling, 196 ‘correlation’ energy and potential, 186 Index ‘exchange correlation’ energy and potential, 186 ‘exchange’ energy and potential, 186 electron-interaction energy and potential, 186 local density approximation, 316 of Hartree–Fock theory: physical interpretation, 200 of Hartree theory: physical interpretation, 201 physical interpretation, 185 physical interpretation of ‘correlation’ energy and potential, 199 physical interpretation of electroninteraction energy and potential, 187, 189 physical interpretation of ‘exchange’ energy and potential, 198 physical interpretation of ‘exchangecorrelation’ energy and potential, 189, 190 physical interpretation of Hartree energy and potential, 189 Koopmans’ theorem, 123 L Laplace transform, 402 Local charge distribution, 18 Local density approximation, 11, 345 313, 314 analysis via Quantal density functional theory, 319 Correlation–Kinetic effects, 330 electron-interaction energy according to Q–DFT, 322 ‘exchange’ energy functional according to Kohn–Sham theory, 318 ‘exchange’ potential according to Kohn– Sham theory, 318 ‘exchange–correlation’ energy functional according to Kohn–Sham theory, 316 ‘exchange–correlation’ potential energy according to Kohn–Sham theory, 317 exchange potential according to Q–DFT, 322 Fermi hole, 324 Fermi hole according to Q–DFT, 322, 325 Fermi–Coulomb hole according to Q– DFT, 323 for Kohn–Sham ‘exchange’, 313 Index for Kohn–Sham ‘exchange–correlation’, 313 pair–correlation density according to Kohn–Sham theory, 316 pair–correlation density according to Q– DFT, 321, 323 pair-correlation density according to QDFT, 401 Pauli field according to Q–DFT, 325 S system differential equation, 317 wavefunction, 323 within Slater theory, 338 Local density functional theory exchange potential according to Q–DFT, 328 Local effective potential energy theory, 2, Local electron-interaction potential energy, Local potential energy operator, Lorentz field, 253, 284, 285 Lorentz force equation, 258 Low-electron-correlation regime, M Magnetization density, 274 Magneto-caloric effect, 256, 284 Magnetoresistance, 256, 284 Marginal probability amplitude, 221 Meissner effect, 256 Metal-oxide-semiconductor structures, 284 Momentum field, 35 N Newton’s laws, 27, 32 Non conserved energy, 17 Non-conserved total energy, Noninteracting bosons, 6, 216 Noninteracting fermions, 3, 161 Noninteracting v–representability, 159 Nonlocal charge distribution, 21 N -representable densities, Nuclear magnetic resonance, 256 O Operator canonical angular momentum, 263 canonical kinetic energy, 260 canonical momentum, 260 current density, 22 density, 18 diamagnetic component, 261 411 Hermitian, 18–20, 22 momentum, 151, 168 magnetization current density, 277 multiplicative or local, 68 pair–correlation, 20 paramagnetic component, 261 physical angular momentum, 264 physical current density, 260, 261 physical kinetic energy, 261 physical kinetic energy in magnetic field, 259 physical momentum, 260 single-particle density matrix, 18 translation, 19 unitary, 149, 168 Optimized Potential Method (OPM), 11, 186, 202, 345 ‘exchange–only’ approximation, 203, 250 ‘exchange-only’ version, 203 physical interpretation, 186, 208 physical interpretation of ‘exchangeonly’ version, 186, 208 Orbital–dependent ‘exchange potential energy’, 122 Orbital–dependent Fermi hole, 122 P Pair–correlation density, 17, 19, 22, 72, 74, 349 Pair–correlation function, 21, 74, 288 Pair function, 20 Paramagnetic current density, Pauli correlations, Pauli–Coulomb energy, 55 Pauli–Coulomb field, 22, 53 Pauli energy, 95 Pauli exclusion principle, 3, 16, 19, 46 Pauli field, 75 Pauli kinetic energy, 6, 217 density functional theory definition, 220, 221 quantal density functional theory, 229 Pauli potential, Pauli potential energy, 217, 220 density functional theory definition, 221 quantal density functional theory definition, 228 Percus-levy-lieb theory, 9, 155, 255, 343 for noninteracting fermions, 159 Physical angular momentum, 259, 264 Physical current density, 284, 288 Poynting’s vector, 257 412 Q Quantal compression, Quantal compression of kinetic energy density, 116 Quantal decompression, Quantal decompression of kinetic energy density, 116 Quantal density functional theory, 68, 341, 343 of Hartree theory, 117 expression for the discontinuity, 243 generalization to external electrostatic and magnetostatic fields, 284, 291 in terms of adiabatic coupling constant perturbation theory, 196 of degenerate states, 113 of Hartree–Fock theory, 117, 123 of Hartree theory, 129 of the density amplitude, 215 of the discontinuity, 232 of the discontinuity of the electroninteraction potential, 239 Pauli–correlated approximation, 123, 208, 247, 324, 336 time-dependent, 71, 89 time–independent, 91 within adiabatic coupling constant scheme, 192 Quantal density functional theory of the B system electron–interaction energy, 226 electron–interaction field, 225 ‘Quantal Newtonian’ laws first law, 4, 38, 353 second law, 28, 342 first law in the added presence of a magnetic field, 284, 287 first law in the presence of a magnetic field, 253, 383, 389 second law, 2, 27, 349, 352 Quantal sources, 4, 15, 17 Coulomb hole, 93 Dirac density matrix, 93 electron density, 18, 93 Fermi–Coulomb hole, 93 pair–correlation density, 93 Quantal torque–angular momentum equation, 35 Quantum dot, 295, 284 Quantum fluid dynamics, 8, 16, 59 Quantum Hall effect, 256, 284 Quantum–mechanical ‘hydrodynamical’ equations, 31 Index Quantum wells, 284 R (v, A)-representable densities, 254 Runge–Gross theory, 2, 164, 343 corollary, 9, 178 action functional, 165 ‘correlation’ action functional, 167 correlation–current–density effects, 166 correlation–kinetic effects, 166 electron-interaction action functional, 166 ‘exchange’ action functional, 167 ‘exchange–correlation’ action functional, 167 generalization, 167 Keldysh action, 166 van Leeuwen theorem, 166 S Scalar potential, 2, 284 Schrödinger-Pauli theory, 7, 273 Hamiltonian, Schrödinger theory, 1, 15, 342 intrinsic self-consistent nature of Schrödinger equation, 8, 29, 39, 342 intrinsic self-consistent nature of Schrödinger equation in a magnetic field, 290 in the presence of a magnetostatic field, 285 time-dependent, 16 time-independent, 38 Self–interaction–correction energy, 128 Self–interaction–correction field, 129 Self–interaction–correction potential, 128 Single–particle density matrix, 21, 349 Single–particle matrix, 19, 288 non idempotency, 19 Slater ‘exchange potential’, 315, 331 Slater theory, 10, 68, 313, 315, 330 X α approximation, 313, 338 for ‘exchange’, 313 local density approximation, 313, 338 Slater exchange potential, 330 Slater potential, 11 Spin angular momentum, 7, 254, 276 Spin density functional theory, 277 Spinless single–particle density matrix, 17 Spinless sources single–particle density matrix, 18 Spinless two–particle density matrix, 349 Index S system, 4, 69 differential virial theorem, 379 differential virial theorem for the timeindependent case, 381 effective field, 82 electron–interaction potential, 85 electron–interaction potential energy, 83 Hartree potential, 85 in the added presence of a magnetic field, 284 internal field, 82 ‘Quantal Newtonian’ second law, 81, 379 torque sum rule, 87 zero force rule, 86 S system (time–independent) energy components in a magnetic field, 294 Fermi hole, 101 fields, 94 ionization potential, 112 pair-correlation density, 100 Quantal Newtonian’ first law in a magnetic field internal field components in an external magnetic field, 292 quantal sources, 93 Sum rules, 86, 97 for Coulomb hole, 75 for Fermi–Coulomb hole, 21 for Fermi hole, 74 for orbital–dependent Fermi hole, 122 for pair–correlation density, 20, 74 for total electron charge, 18 Super lattices, 284 T Time-dependent electric field, 413 Time-dependent external field, Total energy, Two–particle density matrix, 62 U Uniform magnetic field, 254 Unitary transformation, 8, 148, 255 current density preserving, density and physical current density preserving, 262 density preserving, 8, 149, 167 V Valence band, 232 Variational principle for the energy, 119, 127, 142 Vector potential, 2, 151, 168, 284 Velocity field, 35 Von Weizsäcker kinetic energy functional, 220 v-representable densities, 9, 136, 145 W Wigner regime, 114 application of Q-DFT, 114 Y Yrast states, 256 Z Zeeman effect, 256, 284 .. .Quantal Density Functional Theory Viraht Sahni Quantal Density Functional Theory Second Edition 123 Viraht Sahni Brooklyn, NY USA ISBN 978-3-662-49840-8... 53 56 57 57 59 59 60 61 65 Quantal Density Functional Theory 3.1 Time-Dependent Quantal Density Functional Theory: Part I 3.1.1 Quantal Sources ... Time-Dependent Quantal Density Functional Theory: Part II 3.4 Time-Independent Quantal Density Functional Theory 3.4.1 The Interacting System and the ‘Quantal