Development Of A Correction Term For The Kinetic Energy Density Functional
ABSTRACT
Density functional theory (DFT) is a useful theoretical and computational tool for electronic structure calculations, which form the basis for the classification of materials into conductors, semiconductors or insulators. DFT started with a crude approximation by Thomas and Fermi (TF theory) which calculated the kinetic energy of electrons using the socalled local density approximation (LDA). Although TF is computationally inexpensive, it provides a poor numerical result due to a lack of understanding of the density dependence of the kinetic energy. Another approximation to the kinetic energy is the vonWeizsacker (vW) term, which greatly improves the TF theory, yet the full functional form of the kinetic energy remains unknown. We seek to develop a supplemental term to the kinetic energy density functional and compute corrections to the ThomasFermivonWeizsacker kinetic energy of closed shell atoms in order to improve its accuracy.
TABLE OF CONTENTS
1 Introduction
1.1 What is Density Functional Theory (DFT)? . . . . . . . . . . . . . . . . . . 2
1.2 Why DFT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Uses of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Focus of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 The Background: The Schrodinger Equation . . . . . . . . . . . . . . . . . . 4
1.5.1 One Particle TISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.2 System of Several Particles . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.3 A Real System and The BornOppenheimer Approximation . . . . . 6
1.6 The HartreeFock Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 HohenbergKohn Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.1 Representability of Density . . . . . . . . . . . . . . . . . . . . . . . 12
2 Basic Tools 14
2.1 Density Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Uniform Electron Gas (UEG) Model . . . . . . . . . . . . . . . . . . 14
2.1.2 Local Density Approximation (LDA) . . . . . . . . . . . . . . . . . . 14
2.1.3 Gradient Expansion Approximation (GEA) . . . . . . . . . . . . . . . 15
2.2 ThomasFermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The von Weizsacker Functional (vW) . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Conceptualization of Density Matrices . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Electron Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Pair Density: n2(1; 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 Reduced Density Matrix (RDM) . . . . . . . . . . . . . . . . . . . . . 20
iv
3 Calculation of Corrections for KEDF of closed shell systems 21
3.1 Kinetic Energy Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Expression for First Order RDM . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The Kinetic Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Kinetic Energy Corrections for some closed shell Systems . . . . . . . . . . . 26
4 Development of a Correction term and Discussion of Results 28
4.1 Development of a correction term . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Results of the Correction Terms . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bibliography 46
CHAPTER ONE
Introduction
The quantum mechanics of manyelectron systems which have descriptions from time dependent and timeindependent Schrodinger and Liouville equations, is to a good approximation ostensibly a wellunderstood subject. The Schrodinger equations present the theoretical bases for the description of both the time evolution and pure stationary states properties of atoms and molecules. In treating some quantum mechanical systems such as biological molecules and liquids where the individuality of molecules ceases to exist, rather collective erects becomes predominant, it is immaterial to talk of pure states but paramount to consider ensemble of states describable with timedependent and timeindependent Liouville equations in lieu of Schrodinger equations.
However, in each case of pure states and ensemble of nontrivial manyelectron systems, the equations involved are not without complicated and complex mathematical parameters with little or no analytical or numerical solutions. In other words, systems containing thousands of electrons and hundred of nuclei are computationally demanding and, in fact, the problem is such that even if supercomputers were to be improved by several orders of magnitude, both in speed and memory, it would still be difficult, if not impossible, to obtain succulently accurate solutions to these equations [13].
Although, nonrelativistic Hamiltonian operators for systems interacting Coulombically can be written explicitly for these equations, understanding a priori, the subtleties of the many body behavior that ensues from these interactions remains a challenge. Thus, this calls for the formulation of a rigorous quantum mechanical approach entirely equivalent to the Schrodinger or Liouville equations which certainly opened ways for important developments in atomic, molecular and condensed matter physics as well as in quantum chemistry, particularly, to avoid the particlenumber dependency. This particle number dependence, perhaps, had made it impossible to solve the equations for extended and realistic systems. Avoiding particlenumber dependence is a keystone behind the formulation of what has been generically called \density functional theory (DFT)”.
1.1 What is Density Functional Theory (DFT)?
To better understand the properties of materials, and to classify materials so that they can nd application(s) in different spheres of life, one needs to calculate among other properties the electronic ground state structure of the material. From an electronic structure calculations (e.g., band structure calculation), one can classify a material into a conductor, a semiconductor or an insulator.
There are variety of, but not too many variant approaches, approximations, or principles to these calculations, depending on the system. These include; starting form idealized one electron Schrodinger problem to manyelectron problems and then, to real systems; many approximations, principles and theories such as perturbation theory, variational method, the BornOppenheimer approximation, the HartreeFork method, density functional theory; not forgetting the symmetry requirements that employ Pauli’s exclusion principle and the Slater determinant for noninteracting fermions, etc.
Among these approaches, Density Functional theory has been held to high esteem as a linchpin of electronic structure calculation in solid state physics[1], and has made an unparalleled impact on the applications of quantum chemistry which include understanding of electron transport in solar energy materials[2]. This, perhaps, is because analytical solutions of the Schrodinger equations hold for only few simple systems, and numerical exact solutions can be obtained for a small number of atoms and molecules. Again, the recent progress in the calculation of the electronic structure of atoms, molecules and solids has emphasized, mayhap, how far we are from the objective of being able to predict the physical properties of manyelectron systems reliably and with less excessive computation which DFT presents[3]. Thus, density functional theory is somewhat a completely dierent but formally rigorous way of approaching any interacting problem, by mapping it exactly to a much easiertosolve noninteracting problem that is more computationally efficient than HartreeFock; though it has its own problems but, in a nutshell, Definition
1.1.1. DFT is a formalism or a way of simplifying the manybody (particle) problem by working with the electronic charge density as fundamental variable rather than the wave function and trying to nd a direct relation between this density and the energy of the system.
1.2 Why DFT?
The reasons why, at present, density functional theory has become such an attractive theory are really quite clear.
1. It provides reformulation in terms of the oneparticle density, a unique and fundamental quantity which depends only upon three spatial and one spin variables, regardless of the number of particles of the physical system which it seeks to describe.
2. The oneparticle density is an observable and a threedimensional quantity that can be measured experimentally which theorists believe will elucidate the conceptualization of the properties of materials such as the nature of the chemical bond[4].
3. There is much to be gained in terms of the implications that such a reformulation could bring in the numerical handling of quantum mechanical problems.
4. It furnishes interpretative tools which enable researchers to grasp the essential features of physical systems; preferable, to have a vivid picture of the behavior of the wave functions, a simple description of the essence of the factors which determine cohesion, and understanding of the origins in the variations in the properties from metal to metal.
5. The complications and cost associated with orbital manipulations, including orbital orthonormalization and localization are avoided.
6. Particle number dependence of the wave function is avoided
7. DFT has many application in Chemistry and in physics such as calculating the binding energy of molecules in chemistry the band structure of solids in physics.
1.3 Uses of DFT
1. Electronic structure calculation for bandgap, density of states (DOS), and material classification.
2. Determination of the mechanical properties of a material, e.g toughness, bulk modulus.
3. In Chemistry: to predict molecular properties (molecular structure, lattice constant, etc)
4. In the search for new materials with exceptional and novel properties.
5. For molecular dynamics simulation, etc.
1.4 Focus of the Work
Our target in this work is to study the existing kinetic energy density functional of ThomasFermi and vonWeizsacker which has not given exact numerical or analytical solution for systems with more than two electrons. We will compute corrections for the kinetic energy of some closed shell atoms under the restricted HartreeFock scheme, and develop correction terms for the kinetic energy functional that depend on the electron density and its gradient.
Our anticipation is that the correction will be useful for any system ofNelectrons.
1.5 The Background: The Schrodinger Equation
DFT has been in application, historically. The idea as mentioned earlier is to regard the total particle density as the fundamental quantity from which properties of a system can be calculated (determined). So, we here capture some fundamental concepts with some theoretical framework based on Time Independent Schrodinger Equation (TISE). We will
maintain natural units (Hartree atomic unit).
1.5.1 One Particle TISE
Our interest is to develop a correction term for the kinetic energy density functional (KEDF) of many electron closed shell systems. A good point to start is the TISE for a single particle in an external potential v(~r):
^H
(~r) = E (~r) (1.1)
where,
^H
=
1
2
r2 + v(~r) (1.2)
Eq.(1.1) is an eigenvalue equation for the energy operator ^H and denes all possible states of a system, (~r) and their energies (eigenvalues), E. However, when there are disturbance in the system such that the Hamiltonian diers from the ground state Hamiltonian, ^Ho, some form of approximations (perturbation and variational theories) are employed to seek for the ground state properties of the system (Note: we are interested in the ground state properties because that is where the true properties of a system can be explored):
^H
= ^Ho + ^H 0
^H
is the total Hamiltonian, ^Ho is the Hamiltonian for the undisturbed system and ^H 0 is the Hamiltonian resulting from disturbance. is a nonnegative continuous small number 1[18].
Perturbation Theory is a systematic procedure of obtaining approximate solutions to the perturbed problem by building on the known solution of the unperturbed case[19] and is employed when we are able to solve exactly the TISE for the unperturbed case:
^H
o on
= Eon
on
Variational theory states that the expectation value of the energy operator determined from any trial function trial obeying the same boundary condition as the correct wave function of the system cannot be lower than the exact ground state energy Eo or Egs of the system[18] R trial(~r) ^H trial(~r)d3 R ~r
trial(~r) trial(~r)d3~r
Egs (1.3)
The method is employed when we are unable to solve exactly the TISE for the unperturbed case of the the system and therefore are looking for approximate solution.
1.5.2 System of Several Particles
The Hamiltonian operator for N manyparticle system with zero order approximation (turn o electronelectron interaction) is:
^H
=
XN
i=1
1
2
r2i + V (~ri ~rN) (1.4)
The wave function is a very complicated function of the coordinates of the particles and is given in what is known as orbital approximation[20]. In orbital approximation, the reasonable approximation to the exact wave functions is obtained by thinking of each particle as occupying its “own” orbital, i.e as a pure state describable by a wave function then, the wave function of the ensemble or mixed state as the product of single particle wave function:
(~r) = (~r1; ~r2; ; ~rN) = (~r1)(~r2) (~rN) (1.5)
If all the particles are of the same kind then (~r) must satisfy some special symmetry properties since the expectation value of any operator, ^O
^O
=
Z
(~r1; ; ~rN)^O (~r1; ; ~rN)d3~r1 d3~rN (1.6)
of a particular observable of a system must be invariant under the interchange of any two identical particle’s coordinates, ~rj and ~rk
^O
=
Z
(~r1; ; ~rj ; ; ~rk; )^O (~r1; ; ~rj ~rk )d3~r1 d3~rj ~rk
=
Z
(~r1; ; ~rk; ; ~rj ; )^O (~r1; ; ~rk ~rj )d3~r1 d3~rk ~rj (1.7)
This is true on the condition that the probability of finding a particle irrespective of interchange of coordinates remains the same:
(~rj ; ~rk) (~rj ; ~rk) = (~rk; ~rj) (~rk; ~rj) =) j (~rj ; ~rk)j2 = j (~rk; ~rj)j2 (1.8)
The necessary requirements for eqs (1.7) and (1.8) is
(~rj ; ~rk) = (~rk; ~rj)
1. (~rj ; ~rk) = + (~rk; ~rj) =) symmetric and this is satised by bosons, and
2. (~rj ; ~rk) = (~rk; ~rj) =) antisymmetric and it is satised by fermions (e.g. electrons)
In the case of just two particles, the two particle wave function (~x1; ~x2) formed by taking product of single orbital wave function of the individual particles is:
HP (~x1; ~x2) = a(~x1)b(~x2)
Here, ~x consists of the spatial coordinate ~r and the spin coordinate s. So that i(~x) = i(~r; s); i = a; b: But if the particle are electrons of the same spin, the antisymmetric requirement is not satised since (~x1; ~x2) 6= (~x2; ~x1): However, a twoelectron orbital wave function that satises the antisymmetric condition can be formed by adding a second term that is negative of the rst term with the coordinates labels interchanged[18].
(~x1; ~x2) =
1
p
2
[a(~x1)b(~x2) a(~x2)b(~x1)] (1.9)
The factor p1
2
is to ensure normalization. We observe that (~x1; ~x2) = (~x2; ~x1) and if the spin orbitals a(~x1) and b(~x2) are the same, that is, if a = b then, (~x1; ~x2) = 0 8 ~x1; ~x2.
In this way, the Pauli exclusion principle (PEP) that no two electrons can occupy the same spinorbital is obeyed. A general way of forming an antisymmetric wave function from N noninteracting system of electrons is given by Slater determinant:
(~x1; ~x2; ; ~xN) SD =
1
p
N!
1(~x1) 2(~x1) N(~x1)
1(~x2) 2(x2) N(~x2)



1(~xN) 2(~xN) N(~xN)
(1.10)
1.5.3 A Real System and The BornOppenheimer Approximation
In the manyelectrons system considered, we neglected the repulsive interactions between the electrons. However, this is an idealization because, it is not true for most practical real systems. In real systems with very few exceptions, the particles interact. So, the Hamiltonian has more terms in it than in the idealized case, and the wave function is more complicated than what we have seen in a system of noninteracting electrons. The wave function of the kth state of a real system which depends on 3N spatial coordinates f~rkg, and N spin coordinates fskg of the electrons; denoted by f~xkg and the 3M spatial coordinates f~Rkg of the nuclei is:
k(~x; ~R) = k(~x1; ~x2; ; ~xN; ~R1; ~R2; ; ~RM) (1.11)
Then, the TISE for the system is:
^H
(~r; ~R) k(~x; ~R) = Ek k(~x; ~R) (1.12)
6
^H
(~r; ~R) is the Hamiltonian operator for a real system consisting of M nuclei and N electrons and it represents the total energy of the system[16]:
^H
(~r; ~R) =
XN
i=1
1
2
r2i
 {z }
^ Tele(~r)
XM
A=1
1
2MA
r2
A
 {z }
^ Tnuc(~R)
NX;M
i;A=1
ZA
riA
 {z }
^ Vele;nuc(~r;~R)
+
XN
i=1;j>i
1
rij
 {z }
^ Vele;ele(~r)
+
XM
A=1;B>A
ZAZB
RAB
 {z }
^ Vnuc;nuc(~R)
(1.13)
i and j run over the N electrons, whereas A and B run over the M nuclei. ^ Telec and ^ Tnuc are the kinetic energy operators of the electrons and the nuclei respectively. MA is the mass of nucleus A. Vele;ele, Vele;nuc and Vnuc;nuc are the electronelectron repulsive, electronnuclear electrostatic attractive and nuclearnuclear repulsive interaction potential operators respectively.
rij = j~ri ~rj j is the distance between electrons i and j; RAB = j~RA ~RBj is the distance between nuclei A and B, and riA = j~ri ~RAj is the distance between electron i and nucleus A.
In an exact quantum mechanical treatment of a real system, TISE has to be solved for Mnuclei and Nelectrons, but this is practically not feasible. However, the problem is assuaged by the adoption of a number of approximations. The rst approximation to solving the problem is the BornOppenheimer approximation (BOA) also called ClampedNuclei approximation.
Born and Oppenheimer assumed that since the nuclei are much more massive than the electrons, the motion of the electrons are rapid compared with the motion of the nuclei.
Thus, the nuclei can be assumed to be clamped at xed internuclear distances RAB. Hence, the electrons are in motion in the eld of the nuclei whereas the nuclei are in motion in the electronic potential surfaced as Etot[18, 20].
With the BOA, the kinetic energy of the nuclei is zero and the nuclearnuclear repulsive potential energy can be treated as constant for a xed conguration of the nuclei. This is because any constant added to an operator only adds to the operators eigenvalue and has no effect on the operator eigenfunction. Thus, eq.(1.13) reduces to just the electronic Hamiltonian:
^H
(~r; ~R) =
XN
i=1
1
2
r2i
 {z }
^ Tele(~r)
NX;M
i;A=1
ZA
riA
 {z }
^ Vele;nuc(~r;~R)
+
XN
i=1;j>i
1
rij
 {z }
^ Vele;ele(~r)
(1.14)
The TISE for the electronic motions is now given by:
^H
ele ele(~r; ~R) = Eele ele(~r; ~R ) (1.15)
ele(~r; ~R) describes the motion of the electrons and depends explicitly on the electronic coordinates f~rig and parametrically on the nuclear coordinates f~RAg[17]. Parametric de7
Independence means that for dierent arrangement of the nuclei, ele is a dierent function of the electronic coordinate. Although, we did not mention spin which implies we have been dealing on only spatial coordinate, the ele depends on the spinorbital coordinate. To fully describe the wave function, we will employ the choice of Slater determinant discussed earlier.
The energy from the potential of the nuclearnuclear interaction of clamped nuclei is:
Enuc =
XM
A=1
XM
B>A
ZAZB
jRA RBj
(1.16)
Thus, the approximated total energy of a real system of electrons is:
Etot = Eelec + Enuc (1.17)
Through BOA, the electronic Hamiltonian is successfully separated from the nuclear Hamiltonian and the wave function is also separated for every xed value of R as:
(~r; ~R) = (~r) (~R )
The quantity of interest is contained in the electronic Hamiltonian therefore, we will focus on eq.(1.15) and detach the subscript \ele”.
Although, it appears very simple the resulting equation is not a soft nut to chew.
^H
(~r) = E(~r) (1.18)
Its exact solution even for the simplest molecules with two nuclei and an electron remains to the present day a major challenge. So, the quoted words of PaulDirac, 1929 as contained in Von[21] remains the same
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.
Its difficulties lie in the electronelectron interaction term 1
rij which leads to particle (electron) number dependence of the wave function and diculty in writing the wave function of the ensemble and 3N or 4N degrees of freedom of the electrons. As a result, these have called for many approximate methods and theories of solutions
1.6 The HartreeFock Approximation
It is hopeless to anticipate an analytical or numerical solution of eq.(1.18) whose Hamiltonian is eq.(1.14) with such a complicated potential energy terms, however, if an effective potential Veff (~r) can be found for such potential energy term such that a single particle wave function i(~x) that satises one particlelike TISE:
h
1
2
r2 + Veff (~r)
i
i(~x) = ii(~x) (1.19)
is obtained then, computational techniques can be applied to give detailed and reliable numerical result. This is the essence of the HartreeFock (HF) approximation. It replaces the complicated manyelectron problem by a oneelectron problem[17]. This is achieved by reducing the manyelectron Hamiltonian to a singleelectron Hamiltonian with an effective potential:
Veff (~r)i(~r) =
Z
n(~r0)i(~r)
j~r ~r0j
d3~r0
 {z }
Coulomb repulsion
Z
n(~r; ~r0)i(~r0)
j~r ~r0j
d3~r0
 {z }
exchange
XM
A=1
ZAi(~r)
j~r ~RAj
 {z }
external term
(1.20)
The Coulomb repulsion is the electrostatic interaction between two electrons at points ~r and ~r0, which manifests in the eective potential through 1 j~r~r0j and prevents the two electrons from coming too close to each other. The exchange has no classical analogue. It is in no way connected to the charge of the electrons, but it is a direct consequence of Pauli’s exclusion principle[16] which the Slater determinant that denied the HartreeFock product of single particle wave function is satisfied.
Recall that if there are no electronelectron interaction (zero order approximation) in the system, eq.(1.20) reduces to a single particle problem with
Veff (~r) = Vext(~r) =
XM
A=1
ZA
j~r ~RAj
(1.21)
To obtain a oneparticle wave function that satisfy eq.(1.19), D. R. Hartree made an approximation for the wave function by taking the product of N oneelectron orbital wave functions
(~r1; ~r2; ; ~rN) ! 1(~rN) N(~rN)
while V. Fock modified the product by introducing the antisymmetrization property. The antisymmetric product is the HF approximation and it is given by the Slater determinant of eq.(1.10). The choice of Slater determinant guaranteed that Pauli’s Exclusion Principle (PEP) is obeyed. Eq.(1.19) is the HF equation. So, the HF approximation consists of approximating the Nelectron orbitals by an antisymmetric product of N oneelectron wave function i(~x) composed of the spin coordinate functions1(s) and the spatial coordinate functions i(~r) The optimal trial wave function is the one that minimizes the HartreeFock energy according to the variational method
EHF = minfigNi
=1
trial(~x)j[1
2r2 + Veff (~r)]jtrial(~x)
htrial(~x)jtrial(~x)i
EGS (1.22)
figNi
=1 is a set of single particle orbitals. Full minimization of the functional EHF with respect to all allowed Nelectrons wave functions will give the true ground state 0 and energy EHF (0) = EGS. EGS is the true ground state energy.
It necessary to note that the singledeterminant description from the Slater determinant for orbits of manyelectron is an approximation and can never give the exact energy for the manyelectrons system. For higher accuracy in the energy calculations, the exact wave function for a system of many interacting electrons is never a singledeterminant or a simple combination of a few determinants. Owing to variational principle, EHF is necessarily always larger than the exact ground state energy EGS. The difference between these two energies is called correlation energy EHF
c :
EHF
c = EGS EHF 0 (1.23)
EHF
c is a negative quantity since EGS < 0 and EHF < 0, therefore, jEGSj > jEHF j. Thus,
EHF
c is a measure for the error introduced through the HF scheme. Electron correlation is actively caused by instantaneous repulsion of the electrons, which is not covered by the effective HF potential.
The HF method is also called the self consistent eld (scf) method. It involves solving the HF equation by assuming a trial wave function then, the new solution obtained (wave function) becomes the new trial wave function, and the iteration is continued that way until the subsequent iterations produce consistent result with the previous ones.
Expansion of eq.(1.19) for some basis functions, gives the HF energy terms as: Kinetic energy term
Z
p(~r)
1
2
r2q(~r)d~r (1.24)
ElectronNuclear attraction term
Z
p(~r1)
1
r1A
q(~r1)d~r1 (1.25)
e erepulsive energy term
Z Z
p(~r1)q(~r1)
1
r12
a(~r2)
b(~r2)d~r1d~r2 (1.26)
The spin function denoted by (s) is either (s) or (s)
The HartreeFock energy EHF 6= EGS. Its calculation is computer intensive. Density Functional Theory (DFT) is a computational technique employed for solving these equations in order to determine the electronic properties of any system and EGS an the expense of HartreeFock. There are two theorems which establishes DFT.
1.7 HohenbergKohn Theorems
Theorem 1.7.1 (The rst HohenbergKohn theorem). The external potential (~r) is (to within a constant) a unique functional of the ground state electron density n(~r); since, in turn, (~r) xes ^H we see that the complete manybody ground state is a unique functional of the density n(~r)[22]
The theorem is just saying that for an isolated manyelectron system, its groundstate one electron density n(~r) uniquely determines the external potential, and in turn, the density is a functional of the external potential. Thus, the external potential and the density are in a onetoone relation[13, 14, 15, 16]: n(~r) () (~r)
To prove this theorem, we consider two systems each of Nelectrons at the ground state. Suppose there were two external potentials (~r) and 0(~r) not restricted to Coulomb potentials, and that dier not only by a constant. From the two external potential, there are two Hamiltonians ^H = T +Vee +Ven and ^H 0 = T +Vee +V 0
en corresponding to two different ground state eigenfunctions with energy E and 0 with energy E0[15, 16, 22]. HohenbergKohn assumed that the dierent potentials will give dierent Hamiltonians with dierent ground state properties but the same ground state electron density, i.e, n(~r) = n0(~r). This proof uses variational principle therefore, if we take to be a trial wave function for ^H 0
then,
E0 <
D
j ^H 0j
E
=
D
j ^H + ^H 0 ^H j
E
= E + h j0(~r) (~r)j i
=) E0 = E +
Z
n(~r)[0(~r) (~r)]d3~r (1.27)
also taking 0 as a trial wave function for ^H , we obtain
E <
D
0j ^Hj 0
E
=
D
0j ^H 0 + ^H ^H 0j
E
= E0 + h j(~r 0(~r))j i
=) E = E0
Z
n(~r)[0(~r) (~r)]d3~r (1.28)
2So that their kinetic energy T and electronelectron potential Vee(~r) are equal
By adding eqs (1.27) and (1.28) we have:
E0 + E < E + E0 or 0 < 0
which is a contradiction. This implies that there are no two dierent external potential that can give the same ground state electron density. Theorem 1.7.2 (The second HohenbergKohn theorem). The energy E[n] of any vrepresentable
trial density n(~r) places an upper bound to the ground state energy E[no],
i.e,
E[n] E[no] (1.29)
The theorem uses variational principle under the constraints that for some external potential, the trial density satisfy these conditions
• n(~r) 0
•
R
n(~r)d3~r = N
•
R
jrn
1
2 (~r)j2d3~r < 1
Therefore, the theorem is put as: for any trial density n(~r) which satisfies the conditions that n(~r) 0,
R
n(~r)d3~r = N and is associated with some external potential ext(~r), the energy obtained from the functional E[n] = FHK[n] + Ven[n] places an upper bound to the ground state energy EGS, but equals EGS if the true ground state density is the trial[14, 16].
Where FHK[n] = T[n] + Vee[n] is the HohenbergKohn universal energy functional.
1.7.1 Representability of Density
Representability of density is the restriction for density to be eligible in variational principle Definition 1.7.3. Vrepresentability: an electron density is vrepresentable if it stems from an antisymmetric groundstate wave function and its Hamiltonian is associated with some external potential (~r) other than the electronnuclear potential.
Definition 1.7.4. Nrepresentability: an electron density is Nrepresentable if it is the density obtained from some antisymmetric wave function.
Thus, for a given density to be able to determine all the ground state properties of a system, it must be vrepresentable. Therefore, since the wave function determines the density and vice versa, it implies that the density and the wave function are in a onetoone relationship.
Hence, the first theorem can be restated as:
Theorem 1.7.5. First HohenbergKohn theorem: There is a onetoone mapping between groundstate wave functions and vrepresentable electron densities. It is through this unique mapping that a vrepresentable density determines the properties of its associated ground state.
A direct consequence of the former theorem is that if all groundstate properties are functionals of the electron density, then these functionals are denied only for vrepresentable densities. Unfortunately, the conditions for a density to be vrepresentable are not yet completely known, since there are densities in single particle systems that do not come from the ground state wave function of any Vext(~r)[6; 23]
Fortunately, Density Functional Theory can be formulated in a way that only requires the density both in functionals and in variational principle to satisfy the Nrepresentability condition 3. It was shown by Gilbert[24] that any reasonable electron density satisfying the three conditions given under theorem (1.7.2) is Nrepresentable.