Eschrig H. Topology and Geometry for Physics

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occ:

ððdu

jðdu

Þððdu

jðdu



occ:

ððdu

Þðu

jðdu

Þððdu

Þðu

jðdu

ÞÞ



occ:

ððdu

Þðu

jðdu

Þððdu

Þðu

jðdu

ÞÞ



The last relation holds again because of dðu

Þ¼0. The m-sum belongs to

the inserted completeness relation and runs over all bands m, occupied and

unoccupied. However, from the last line of the displayed chain of equations it is

seen that the contributions from the occupied bands m cancel. Together with

ð2pÞ

1

o=ok

¼r

; Eq. 8.101 may now be rewritten as (exercise)

ih

ð2pÞ

occ:

unocc:

ðu

mtk

ntk

Þðu

ntk

 Eju

mtk

Þc:c:

ðe

mtk

 e

ntk

dt rEðtÞ¼

dtJ

ðtÞ: ð8:105Þ

Here, j

¼ep=m is the current operator for one electron (dH

¼ð1=cÞj

 dA)

and the whole expression which scalarly multiplies E is the longitudinal compo-

nent of the conductivity tensor r of the electrons expressed by Kubo’s formula as a

current–current correlation function (see Eq. 8.111). This proves that indeed

(8.101) is the time integral over the current density rE of the electrons, no matter

whether E is a quasi-static applied electric ac ﬁeld of some lattice-commensurable

periodicity or the ﬁeld produced by a shift of a sublattice of nuclei. Of course, to

get the total polarization density of the solid the contribution of the nuclei has to be

added.

In passing comparison of (8.105) with (8.101) shows that in the adiabatic limit

the conductivity of the considered situation is a ground state property; the excited

states do not ﬁgure in (8.101). Recall that a gaped band structure is considered

corresponding to an insulating state. The current in the ground state has much in

common with a supercurrent, it is not connected with a dissipative process.

The obtained results have many more facets [2]. For instance, consider the case

where T is a full period of the time dependence of E: Then, the domain M of

(8.101) may be treated as a four-torus, that is, a cycle, oM ¼ [: Consider the

current component J

in the direction of a

through a section element dk

^ dk

ﬁxed values k

; i ¼ 2; 3 which means considering the closed submanifold (two-

torus) M

¼ T

spanned by k

¼ t=T and k

; 0 k

; k

1 with k



þ 1; l ¼ 0; 1, again with oM

¼ [: (The forms A and F considered below are

scale invariant under rescaling of k

; k

, it makes no difference whether the

8.6 Geometric Phases in Quantum Physics 287

circumferences of tori are taken to be 1 or 2p or any other value.) Ask for the

amount of charge ﬂowing through this section element in one period T of time and

integrated from k

¼ 0 to k

¼ 1, that is, from a face of the k-cell to the opposite

face. It is proportional to

: Locally, that is on any chart of M

; F

was

obtained as the (covariant) derivation of A

¼i

occ:

tr ðU

1

ðk

; k

Uðk

; k

ÞÞdk

; o

¼ o= o k

; which is proportional to the Chern–Simons form of

the ﬁrst Chern character on the principal ﬁber bundle ðP; T

; p; UðNÞÞ; A

2pq

ð1Þ

; cf. (8.70). Here, the ﬁber UðNÞ is the Lie group of global (in r-space)

unitary transformations of the N quantum states per k-value (number of occupied

bands, cf. (8.93, 8.96)). Would the relation F

¼ DA

¼ dA

hold globally on

, then due to Stokes’ theorem this amount of charge would be zero,

; oM

¼ [: (The group UðNÞ is non-Abelian, hence

F¼DA¼dAþi½A; A, compare (8.20) where here the forms have an addi-

tional factor ðiÞ; however, because of the trace in A on has ½A; A ¼ 0.)

Although the curvature form F

¼2pch

(cf. (8.60)) is globally deﬁned on M

on the basis of the Chern–Weil theorem and is related to an observable quantity

, the continuation of the local relation F

¼ dA

¼2i

occ:

tr ðo

ðU

1

ðk

;

ÞÞo

Uðk

; k

ÞÞdk

^ dk

to all of M

is obstructed by the topology of P; the

quantum state may acquire a phase of a multiple of 2p around a cycle since M

not simply connected. The amount of charge transported through the unit cell T

(after integration over the section of the k

; i ¼ 2; 3) may be non-zero, but is

quantized. It is an integer multiple of ðe=jT

jÞa

, compare (8.98). The charge

quantum is proportional to the corresponding ﬁrst Chern number C

(do not

confuse it with the ﬁrst Chern class C

ðEÞ considered in the previous section),

¼

tr o

ðU

1

ðk; k

ÞÞo

Uðk; k

Þðo

$ o



; ð8:106Þ

of the ﬁrst Chern character ch

: It is integer for all values of N of the bundles

ðP; T

; p; UðNÞÞ as was shown after (8.97). This fact was ﬁrst mentioned by

Thouless.

If this charge quantum is not ﬁxed to zero by independent physical reasons (e.g.

mirror symmetry equivalence between a

and a

), then quantized charge per

time period T can be pumped through the unit cell of the crystal, for instance, by

vibrating nuclei driving a charge density wave. Also in agreement with the Chern–

Weil theorem, the curvature form F

=ð2pÞ is expressed by the current–current

correlation function (8.105) and as such is independent of the gauge of the

Phys. Rev. B 27, 6083–6087 (1983).

288 8 Parallelism, Holonomy, Homotopy and (Co)homology

electromagnetic potential and of the choice of the k-dependence of the total phase

of the ground state

occ:

while the local connection form A

=ð2pÞ depends

on those choices.

8.6.4 Orbital Magnetism

The role of surface currents in producing the Landau diamagnetism of a non-

interacting free-electron gas has been known as a paradox in physics over many

decades [12]. On application of a homogeneous magnetic ﬁeld the electrons start to

move around the ﬁeld lines on helical curves whose projections on the plane

perpendicular to the ﬁeld are circular cyclotron orbits with a homogeneous distri-

bution of their centers over this plane so that all their currents mutually cancel. If the

electron system is conﬁned in some volume, then the cyclotron orbits crossing

the boundary of this volume cannot be run through, the electrons are reﬂected at the

surface and start a new orbit with a shifted center determined by the reﬂection

conditions. It is easily seen that this shift of the cyclotron orbits produces a current

tangent to the surface and circulating around the volume opposite to the circulation

of a single complete cyclotron orbit. Classically the cancellation of all these currents

still would persist; according to the well known Pauli–van Leeuwen theorem there

cannot be orbital magnetic polarization in an equilibrium state in classical physics.

Quantization of closed orbits is different for cyclotron orbits in the bulk and surface

orbits around the bulk, it produces a slight difference in favor of the surface current.

It creates a magnetization density in the volume which exactly equals the diamag-

netic polarization of Landau’s diamagnetism, including the superimposed de Haas–

van Alphen oscillations at high magnetic ﬁelds. The free-electron model is of course

an extreme idealization, the existence of such non-dissipative (as long as the applied

static magnetic ﬁeld is present) macroscopic currents on the surface of a sample was

long denied for a realistic metal, and Landau obtained the magnetization from the

thermodynamic potential instead. The non-dissipating edge currents were eventu-

ally observed in laboratory in two-dimensional quantum Hall samples. Meanwhile

there is a large body of literature on the subject, see again [2].

Consider a two-dimensional crystalline sample, for the sake of simplicity with a

square lattice with base vectors a

¼ ae

; a

¼ ae

, in a homogeneous magnetic

ﬁeld B ¼rA in z-direction and an electric ﬁeld E in y direction. Use Landau

gauge A

¼ 0; A

¼ Bx  Ect; B ¼ Be

; E ¼ Ee

, both E and B independent of r:

The Hamiltonian becomes (recall that the electron charge is e)

H ¼

h

i



þi

hc

Bx 

h



þ Uðx; yÞð8:107Þ

where Uðx; yÞ is the two-dimensional crystal potential. This Hamiltonian is not any

more periodic in x-direction. However, if one introduces a magnetic translation

operator

8.6 Geometric Phases in Quantum Physics 289

TðRÞ¼exp iR irðe=hcÞe



; R ¼ n

þ n

; ð8:108Þ

with the commutation rules, easily veriﬁed by direct calculation,

TðRÞ TðR

Þ¼exp ðie=hcÞB  R R

ðÞTðR

ÞTð RÞð8:109Þ

and deﬁnes a superlattice

R ¼ n

þ n

whose base vectors are integer multi-

ples of a

and a

, respectively, and such that ðe=hcÞB  a

 a

¼ 2p; then there is

a common eigenfunction system of the Hamiltonian and the magnetic translations

Tð

RÞ¼e

R

¼ u

; H

h

j þ jÞ

þ U;

j ¼irþðe=hcÞe

ðBx EctÞ:

ð8:110Þ

Although the relation (8.109) is most easily directly calculated for a square lattice,

it is written already in a form which holds for any two-dimensional lattice with

base vectors a

; a

. The unit cell of the superlattice of vectors

R is chosen such

that it is penetrated by one quantum U

¼ B 



¼ 2phc=e of magnetic ﬂux.

The eigenvalues j of the operator

j run through the unit cell T

of the lattice

reciprocal to the superlattice.

Kubo’s formula for the static conductivity tensor of the considered situation is

ih

ð2pÞ

occ:

unocc:

ðu

Þðu

Þc:c:

ðe

 e

ð8:111Þ

(Its derivation is beyond the scope of this text.

) Here, a and b are Cartesian

indices, and we consider our two-dimensional case. This expression yields r

¼ 0

(related to constant E, the Fermi level is in a gap between Landau levels) and

¼ r

, the Hall conductivity. Here, as previously, d

j ¼ðð2pÞ

=jT

jÞdj

(the j

run from 0 through 1). Using again j

¼ðe=hÞ½r

; H

 and

¼ð2pÞ

1

o=oj

; j



j¼jT

j, r

may be cast into

2ph

occ:

ð2iÞðo

Þdj

^ dj

; o

: ð8:112Þ

The integrand is again the curvature form F of the local Chern–Simons connection

form A¼

occ:

ðiÞðu

Þdj

on the bundle ðP; T

; p; UðNÞÞ over the closed

manifold T

, and the integral expression in large parentheses is an integer, the ﬁrst

Chern number C

, which due to the Chern–Weil theorem only depends on the

bundle P and is independent of the actually chosen local connection form A, that

See for instance D. Cohen, Phys. Rev. B 68, 155303-1–15 (2003).

290 8 Parallelism, Holonomy, Homotopy and (Co)homology

is, of the gauge of the electromagnetic potential A

which determines the

r-dependent phases of the u

, and of the choice of (smoothly j-dependent)

r-independent unitary transformations of the u

. Hence, r

is quantized,

2ph

; C

2 Z: ð 8:113Þ

On this Chern number form of the Hall conductivity of a two-dimensional electron

system the whole theory of the quantum Hall effect is essentially based. Note that

demands of smoothness in the last expression again require an excitation gap of

eigenenergies e

between occupied and unoccupied states. Here, this is the gap

between adjacent Landau levels, which latter form a discrete sequence in two

dimensions (but not in three). The chosen relation between the magnetic ﬁeld B

and the superlattice just ensures that the Fermi level lies in one such gap. The

corresponding values of magnetic ﬁeld strength would form a discrete sequence

for an ideal two-dimensional crystal, and the corresponding values of the Hall

voltage as function of the ﬁeld strength would just lie on the classical straight line.

However, potential perturbations present in a real sample may ﬁll the gaps

between Landau levels with disorder-localized states, still leaving a mobility gap.

Since the role of the gap in the topological argument is essentially to localize the

states, the Hall conductivity remains quantized (with an extremely high precision

of less than one in 10

) over some interval of magnetic ﬁeld and the Hall voltage

shows plateaus.

Sometimes the situation, although with N occupied bands, is interpreted with

the Uð1Þ-bundle ðP; T

; p; Uð1ÞÞ. This is possible since only the trace of the UðNÞ

group element enters the ﬁrst Chern number which is just a phase.

In the ð2 þ1Þ-dimensional effective ﬁeld theory of the situation,

¼ðct; x; yÞ; l ¼ 0; 1; 2, the transport equation is

¼ r

lmr

012

eff

; S

eff

x d

lmr

012

; o

; ð8:114Þ

where j

¼ cq

is given by the charge density q

, A

is the potential of the elec-

tromagnetic ﬁeld F

¼ d

lmr

012

(U(1) gauge ﬁeld), and the interaction part of the

effective action is the Chern–Simons action S

eff

. (Do not confuse A

and F

with

the forms A and F above.) The settings before (8.107) yield j

¼ cq

B ¼ r

; j

¼ r

E ¼ r

; j

¼ 0. Under the action of an electric ﬁeld in y-

direction there is a Hall current in x-direction; the current does not gain energy out

of the ﬁeld, hence it does also not dissipate energy. The proportionality between the

charge density q

and the magnetic ﬁeld strength perpendicular to the sample might

seem strange; however, the time derivative of this relation together with the con-

tinuity relation for the charge density (charge conservation) yields ð1=cÞ oB

=ot ¼

1

=ot ¼r

1

¼ ðr EÞ

which is Faraday’s law of induction.

So far the analysis (8.108–8.113) was based on the consideration of an inﬁnite

periodic two-dimensional crystal which does not have edges. Consider as a more

8.6 Geometric Phases in Quantum Physics 291

realistic situation a strip extending in y-direction and conﬁned in x-direction.

Assume it of length L in y-direction and closed to a loop (periodic boundary

conditions in y-direction). Choose this direction to be that of

so that periodicity

of the Hamiltonian in this direction is preserved and j

¼ j remains a valid

quantum number of stationary states. In x-direction the states u

are now localized

across the strip with discrete quantum numbers included in n. The Hamiltonian

(8.110) is changed into a one-dimensional Hamiltonian H

; j ¼ jðxÞ¼

io=oy þðe=hcÞðBx  EctÞ. For simple tight-binding models this Hamiltonian

can directly be solved for E ¼ 0. There are still states sufﬁciently far from the

edges of the strip grouped in Landau levels. Close to the edges there appear states

whose energy as function of j disperses across the gaps between Landau levels

in a chiral manner; on the edge with larger x-coordinate their group velocity

oe=oj points in the negative y-direction only and vice versa. The corresponding

edge currents in opposite directions on opposite edges do not lead to a net current

along the strip nor do they dissipate energy. The current in x-direction cannot

stationarily ﬂow any more. Instead it polarizes the sample until the Hall voltage

due to this polarization counterbalances the current driving force due to the voltage

along the sample. Instead of applying an electric ﬁeld E

the magnetic ﬁeld

strength B ¼ B

could adiabatically be changed in time which itself produces an

electric ﬁeld E

¼ðx=cÞoB=ot (in agreement with Faraday’s law). Hence,

=ot ¼oj

=ox ¼r

oE=o x ¼ðr

=cÞoB=ot. The magnetic ﬂux through

the strip is L

B where L

and L

¼ L are the extensions of the strip. The total

amount Q

of charge which ﬂows across the strip is given by dQ

=dt ¼

=ot ¼ðr

=cÞL

oB=ot. Hence a change of ﬂux by one ﬂux unit U

transports precisely one charge unit across the strip polarizing it and contributing

to the Hall voltage (Laughlin). Needless to say that this is only valid as long as the

magnetic ﬁeld varies within one quantum Hall plateau, because the whole

approach is based on the position of the Fermi level in a mobility gap inside the

strip.

This result also provides a connection with the situation of (8.101). Let the

change in magnetic ﬁeld DB be so weak that the variation of DBx through one unit

cell can be neglected. (In semiconductor physics this is related to a switch from

description with a microscopic wave function to one with envelope wave functions

which are averaged over unit cells.) Small changes of E and B may then be treated

as an adiabatic parameter kðx; tÞ on which the Hamiltonian H

kðt;xÞj

and the

quantum state u

ntxj

depend, which, however, can be taken out of the matrix ele-

ments ðu

mtxj

ntxj

Þ. One has ½ðo=otÞ; H

¼ðx=cÞðoB=otÞðe=mÞp

¼ðx=cÞ

ðoB=otÞj

. The polarization in x-direction is given like in (8.105) by dP

=dt ¼ j

hence oj

=ox ¼oq=ot ¼ o

=ðotoxÞ, and with a notation ðt; x; jÞ¼ðq

; q

one has

¼d

; i; j ¼ 0; 1; ð8:115Þ

and with

292 8 Parallelism, Holonomy, Homotopy and (Co)homology

occ:

ðiÞðu

Þ; F¼dA; o

; A ¼ 0; 1; 2; ð8:116Þ

one ﬁnds j

¼1=ð2pÞ

q d

012

i2j

, in particular

¼ j

q ð2iÞðo

Þjo

: ð8:117Þ

The t-integral of which compares precisely with a one-dimensional version of

(8.101), if one identiﬁes ðx=cÞoB=ok with the driving force polarizing the system.

Cycling this force provides another mean to pump charge through the insulator in

the ground state. The ð2 þ1Þ-dimensional quantum ﬁeld theory of the quantum

Hall system reduces to a ð1 þ 1Þ-dimensional quasi-classical ﬁeld theory for an

adiabatic charge pumping process by replacing one wave-vector component by an

adiabatically time-dependent parameter performing a closed loop in parameter

space.

The whole story may be cast in yet another form relevant in most topical types

of solids christened topological insulators, Chern insulators or topological semi-

conductors. Deﬁne phase space variables q ¼ðq

Þ¼ðt; x

; x

; k

Þ;

A ¼ 0; 1; 2; 3; 4; o

¼ o= o q

, the corresponding electromagnetic potential ðA

Þ¼

ðA

; A

; 0; 0Þ and the Berry–Simon connection ððiÞ

occ:

ðu

ÞÞ ¼ ðA

Þ¼ð0; 0; 0; A

; A

Þ. Then, inserting (8.112) in the form

=ðhð2pÞ

ðo

 o

Þdq

for r

, the effective action (8.114) may be

expressed as

ð2þ1Þ

eff

h

2!ð2pÞ

q d

ABCDE

01234

tr ðo

Þð8:118Þ

Now, replace k

þ A

by the adiabatic parameter kðt; x

Þ, replace o

by o

, A

ðiÞðu

Þ by A

¼ðiÞðu

mkq

Þ, remove the integrations over

¼ dx

and over dq

=ð2pÞ¼dk

=ð2pÞ and let the only non-constant gauge

potential A

depend on t; x

¼ q

; q

only, so that d

ABC

012

¼ 2ðA



Þ¼2ðA

k  A

kÞ. This changes (8.118) into

h

dtdx

ðA

k  A

kÞ

tr ðo

 o

Þ:

Now, u

nkq

depends on t and on x

through k only, hence ðo

kÞo

¼ o

and

ðo

kÞo

¼ o

as well as ðo

kÞo

¼ o

and ðo

kÞo

¼ o

which rewrites the displayed expression as

h

dtdx

tr ðo

 o

ÞA

tr ðo

 o

ÞðÞ

or ﬁnally, renaming t; x

; k

into q

; q

8.6 Geometric Phases in Quantum Physics 293

ð1þ1Þ

eff

h

q d

ABC

012

tr ðo

Þð8:119Þ

which is the phase space expression for the interaction part of the action in ð 1 þ 1Þ

dimensions describing the quantum Hall strip, obtained by the dimension reduction

procedure introduced by Qi et al. [13].

Only very recently activities were started to develop a Berry curvature theory of

the orbital magnetic moment density in analogy to theory of the electric dipole

density (see [14]). As mentioned above for the model of free electrons, it is

produced by electric ring currents ﬂowing in the bulk. This time, two dimensions

are at least needed to consider ring currents (Fig. 8.5). While again the average

physical moment density cannot be determined from the current density, it cor-

responds to a counterclockwise ring current density for a periodicity volume which

is an integer multiple of the cells drawn in full lines on the left part of Fig. 8.5, and

to a clockwise ring current for a periodicity volume which is an integer multiple of

the dashed cell, the presence of a surface (not its shape) determines what are the

physical ring currents adding up to a physically measurable edge current along the

boundary of the sample volume.

8.6.5 Topological Insulators

This is a rapidly developing subject which was initiated in the ﬁrst years of the

present millennium. The aspects relevant in our context are best described in [13].

Consider the principal bundle ðP; T

; p; UðNÞÞ over the 2r-torus. The rth Chern

number C

of the rth Chern character is (cf. 8.61), iF

replaced by F

from above)

^^



¼ tr

k ðF



ð2pÞ

tr ðF

ðF

ðF

2r 1;2r



^ dk

^^dk

r!2

ð2pÞ

k d

122r

i

tr ðF

ðF

ðF

2r1



ð8:120Þ

Fig. 8.5 Ring currents with

respect to different

periodicity volumes

294 8 Parallelism, Holonomy, Homotopy and (Co)homology

where ðF

¼ðiÞðo

Þþi½A; A

with A

¼ðiÞðU

1

ðkÞo

UðkÞÞ

. Under the trace the matrices

k F

may be trigonalized without

changing the trace; the skew-symmetric part i½A; A

thereby remaining skew-

symmetric and hence with zero diagonal. The trace of the rth power of triangular

matrices equals the sum of the rth powers of the diagonal elements. Since we found

earlier that tr ð2=2pÞ

kðF



is an integer independent of the order of the

matrix, all diagonal elements of this matrix are integers. Hence, C

is also an integer.

Consider formally the ð2r þ 1Þ-dimensional effective Chern–Simons gauge

ﬁeld theory with the interaction part of the action

ð2rþ1Þ

eff

h

ð2pÞ

o

2r 1

dt ^ dx

^^dx

h

ðr þ 1Þ!ð2pÞ

2rþ1

x d

lm...rs

01...2r

o

;

h

r!ð2pÞ

lm...rs

01...2r

o

ð8:121Þ

Greek indices like l run through 0; 1; ...; 2r. In total as previously for r ¼ 1 the

action integral may be written as

ð2rþ1Þ

eff

h

r!ðr þ 1Þ!ð2pÞ

4rþ 1

q d

...A

01...4r

 A

o

2r1

tr D

2rþ1

2rþ2

D

4r1



ð8:122Þ

with q ¼ðq

Þ¼ðt; x

; ...; x

; k

; ...; k

Þ and the subscripts A

running from 0

through 4r. Now, ðA

¼ðiÞ

occ:

ðu

Þ and DA¼dAþi½A; A.This

is called a ð2r þ 1Þ-dimensional effective Chern–Simons ﬁeld theory for a fun-

damental 2r-dimensional lattice-periodic topological insulator with a Brillouin

torus T

So far there is no experimental realization for r [ 1. However, a dimension

reduction of the ð4 þ1Þ-dimensional case yields a quasi-classical adiabatic

ð3 þ 1Þ-dimensional theory of a three-dimensional topological insulator where in

addition an F

time-reversal symmetry plays a crucial role. Very recently, Bi

, and Sb

have been found as realizations of this case. A further

dimension reduction leads to a ð2 þ1Þ-dimensional theory for a so called spin-

Hall insulator, of which a HgTe quantum dot on CdTe is a realization. For more

details see for instance [15].

The crux in all these cases is that crystal periodicity or periodic boundary

conditions produce tori which on the one hand eliminate surface effects and

preserve volume properties and on the other hand have non-trivial Chern numbers

expressing those volume properties.

In order not to digress to far from our subject we used a Kohn–Sham approach

in the whole presentation. In a more general many-body approach the same sort of

8.6 Geometric Phases in Quantum Physics 295

theory may be formulated with quasi-particle Green’s functions G jÞðj=ðe  RÞ

with a self-energy R, and where Kubo’s formula takes on the form of an

expression tr ðGj

Þ and the (Berry) Chern–Simons form is tr ðG

1

dGÞ:

Otherwise all results remain the same.

8.7 Gauge Field Theory of Molecular Physics

In the Born–Oppenheimer adiabatic treatment of molecular motion and chemical

reactions, the 3N coordinates R ¼ðR

; ...; R

Þ of the N involved nuclear positions

form the adiabatic parameter manifold M for the electronic states jW

ðRÞi. The

adiabatic Hamiltonian is (h ¼jej¼m

¼ 1, with electron charge e and mass m

is the charge of nucleus n)

ðRÞ¼

 r



 r

 R

: ð8:123Þ

Via

ðRÞhrjW

ðRÞi ¼ hrjW

ðRÞiE

ðRÞð8:124Þ

it yields the effective adiabatic potential E

ðRÞ for the nuclear motion:

Hhr; RjW

i¼hr; RjW

iE; hr; RjW

i¼

hrjW

ðRÞihW

ðRÞjhRjW

i: ð8:125Þ

H ¼

=ð2M

Þþ

ðRÞ is still the full Hamiltonian, hrjW

ðRÞi is an r-

dependent wave function parametrically also depending on R, and

ðRÞjhRjW

i¼W

ðRÞ is a nuclear wave function depending only on R while

the r-dependence is integrated out in the Hilbert scalar product. (hrj and hRj are

position eigenstates.) So far (8.125) does not contain approximations.

In order to have a discrete spectrum of

H, the center of gravity of the nuclei is

now eliminated, and further on R denotes relative nuclear coordinates rescaled in

such a way that all corresponding relative masses l are equal. This also implies

neglecting the electron mass in the relative motion. Then, for the ð3N  3Þ-

dimensional Hilbert-space momentum vector operator



P corresponding to the

relative coordinates and obeying hr; Rj



PjW

i¼ir

hr; RjW

i, where r

means

the covariant differential (7.28) in the R-manifold, one easily obtains

ðRÞjhRj



PjW

i¼

id

 iA

ðRÞ



ðRÞð8:126Þ

with

ðRÞ¼hW

ðRÞjr

ðRÞi: ð8:127Þ

296 8 Parallelism, Holonomy, Homotopy and (Co)homology