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Degenerate

ground

states

and

nonunique

potentials:

breakdown

and

restoration

of

density

functionals

K.

Capelle,

1

C.

A.

Ullrich,

2

and

G.

Vignale

2

1

Departamento

de

F´

ısica

e

Inform´

a

tica,

Instituto

de

F´

ısica

de

S˜

a

o

Carlos,

Universidade

de

S˜

a

o

Paulo,

Caixa

Postal

369,

13560-970

S˜

a

o

Carlos,

SP,

Brazil

2

Department

of

Physics

and

Astronomy,

University

of

Missouri-Columbia,

Columbia,

Missouri

65211,

USA

(Dated:

February

6,

2008)

The

Hohenberg-Kohn

(HK)

theorem

is

one

of

the

most

fundamental

theorems

of

quantum

mechan-

ics,

and

constitutes

the

basis

for

the

very

successful

density-functional

approach

to

inhomogeneous

interacting

many-particle

systems.

Here

we

show

that

in

formulations

of

density-functional

theory

(DFT)

that

employ

more

than

one

density

variable,

applied

to

systems

with

a

degenerate

ground

state,

there

is

a

subtle

loophole

in

the

HK

theorem,

as

all

mappings

between

densities,

wave

functions

and

potentials

can

break

down.

Two

weaker

theorems

which

we

prove

here,

the

joint-degeneracy

theorem

and

the

internal-energy

theorem

,

restore

the

internal,

total

and

exchange-correlation

en-

ergy

functionals

to

the

extent

needed

in

applications

of

DFT

to

atomic,

molecular

and

solid-state

physics

and

quantum

chemistry.

The

joint-degeneracy

theorem

constrains

the

nature

of

possible

degeneracies

in

general

many-body

systems.

Introduction.

Quantum

mechanics

is

based

on

the

as-

sumption

that

all

information

that

one

can,

in

principle,

extract

from

a

system

in

a

pure

state

at

zero

temperature

is

contained

in

its

wave

function.

In

nonrelativistic

quan-

tum

mechanics

the

wave

function

obeys

Schr¨

o

dinger’s

equation,

1

which

implies

a

powerful

variational

principle

according

to

which

the

ground-state

wave

function

min-

imizes

the

expectation

value

of

the

Hamiltonian.

This

variational

principle

was

used

by

Hohenberg

and

Kohn

(HK)

2

to

show

that

the

entire

information

contained

in

the

wave

function

is

also

contained

in

the

system’s

ground-state

particle

density

n

(

r

).

HK

established

the

existence

of

two

mappings,

v

(

r

)

1

⇐

⇒

Ψ(

r

1

,...

r

N

)

2

⇐

⇒

n

(

r

)

,

(1)

where

the

first

guarantees

that

the

single-particle

poten-

tial

is

a

unique

functional

of

the

wave

function,

v

[Ψ],

and

the

second

implies

that

the

ground-state

wave

function

is

a

unique

functional

of

the

ground-state

density,

Ψ[

n

].

Taken

together,

both

mappings

are

encapsulated

in

the

single

statement

that

the

single-particle

potential

is

a

unique

density

functional

v

[

n

].

In

this

formulation,

the

HK

theorem

forms

the

basis

of

the

spectacularly

success-

ful

approach

to

many-body

physics,

electronic-structure

theory

and

quantum

chemistry

that

became

known

as

density-functional

theory

(DFT).

3,4,5

Mapping

2

was

originally

proven

by

contradiction

2

and

later

by

constrained

search.

6

Note

that,

in

spite

of

occa-

sional

statements

to

the

contrary

in

the

literature,

nei-

ther

proof

directly

proves

the

combined

mapping,

and

thus

the

existence

of

the

functional

v

[

n

].

This

requires

additionally

mapping

1,

which

in

the

case

of

density-only

DFT

is

proven

by

inverting

Schr¨

o

dinger’s

equation

4,7

ˆ

V

=

󰀁

i

v

(

r

i

)

=

E

k

−

(

ˆ

T

+

ˆ

U

)Ψ

k

(

r

1

,...

r

N