Symmetry properties in position and momentum space

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Symmetry properties in position and momentum space

M. Defranceschi, G. Berthier

To cite this version:

2791

Symmetry

properties

in

_position

and

momentum

_space

M. Defranceschi

₍₁₎

and G. Berthier

₍₂₎

(1)

CEN

_Saclay,

Bât. 462,

DSM/DPhG/PAS,

F-91191 Gif-sur-Yvette, France

(2)

Institut de

_{Biologie Physico-chimique,}

13 rue Pierre et Marie _Curie, F-75005 Paris, France

(Received 19 March 1990,

accepted

in

_{final form}

2 _August

₁₉₉₀₎

Abstract. 2014 Atomic and molecular

symmetry

properties

for the wave functions and electronic densities in

_position

and momentum spaces are

_compared.

A

_{systematic study}

of the

correspon-dance of

_point

groups in both spaces is made and

examples

of

_experiments

of _(e,

_2e)

_spectroscopy

performed

on atoms or molecules are

_{compiled. Simple guidelines}

for

_interpreting

electron distribution _patterns are

_presented

which enable one to sketch the nuclear geometry from the

knowledge

of momentum _{maps and Fourier transform} effects.

J. Phys. France 51

₍₁₉₉₀₎

2791-2800 15 DECEMBRE 1990,

Classification

Physics

Abstracts 32.90 - 02.20

Introduction.

The _symmetry

_properties

of a molecular

_system

with _respect to

_spatial

transformations has

long

been

recognized

to be an

_important

_aspect of _quantum mechanics of molecules and chemical

understanding

often relies on _symmetry considerations. For

instance,

molecular electron

_density

_plots

_provide

a visual

_description

of the electronic structure of molecules

giving

easier

interpretation

of their

_physical

and chemical

_{properties...,}

because of their

conformity

with our usual

physical perceptions.

In momentum _space, such

plots

provide

a

different and _very unfamiliar

_description

of the electronic

_density

of molecules and their

difficult

_{interpretation}

has

_prevented

chemists from

becoming

familiar with this

viewpoint.

Recently

a method

((e, 2e)

spectroscopy)

for

_picking

out one-electron distributions in

momentum space has been

developed.

Unlike

_{scattering techniques}

which

_sample

the

momentum distribution due to all electrons of the target,

binary (e, 2e) experiments

measure

the

_{spherically averaged}

momentum distribution of a

_single

electron in an atom or a molecule

and

_yield

direct measurements of individual electron momentum distributions of the valence orbitals as selected

_by

their ionization

_energies.

Therefore the theoretical

problem

to be solved is how to determine the momentum

wavefunction, 0 (p),

from the

experimental

knowledge

of the momentum

_density,

p

(p ).

The main

problem

is the lack of

_phase

information involved in p

(p).

The purpose of this

paper is to

_investigate

the _{symmetry relations}

_(related

to the

_phase

of the

_complex

wavefunction)

between

_position

and momentum wavefunctions and densities and to correlate

both

_position

and momentum

_point

_groups of these

_quantities

and

_finally

to examine some

important

features that

_might

be

_interesting

for

experimentalists.

A table

gives

the correlation

between momentum and

_position

_space

_point

_groups illustrated

_{by experimental examples ;}

the paper is

deliberately

limited to

_point

_{groups and does} not treat _{space groups} useful in

crystallography.

Numerous _papers referenced

_{by Epstein}

up to 1975

_[1]

and more

_{recently by}

Brion

_[2]

or

_by

Silver and Sokol

_[3]

and

_starting

with the

_early

works of Coulson

[4]

and

_McWeeny

_[5]

have been written both

_by

theoreticians and

_{experimentalists}

on momentum

_{distributions ;}

for

instance,

Kaijser

and Smith

_{[6, 7]}

studied the existence of the inversion center in momentum

space

distributions,

and Tanner

[8]

the bond directional

principle.

Brion et al.

[2]

were

interested in the

_experimental

_aspects, and of course, all the references cited in this paper are

concerned with these

_problems.

Even if numerous _papers have

_already

been

published

on molecular momentum distribu-tions none of them is an extensive review of the theoretical and

_experimental

_symmetry

aspects.

This paper is a _survey of

_{symmetry aspects}

in momentum _space,

_together

with a

tabular

presentation

of

_position

and momentum _space

_point

_groups.

The

_relationship

between momentum and

position

densities for

large

molecules and solids is not included in this _paper.

However,

the reader interested in a wider

_{investigation}

of

momentum distribution in extended _{systems may} have a look at the papers of

Rawlings

and Davidson

_[9],

at those of

_{Mijnarends et}

al.

_{[10, 11],}

Coulson

_[12],

Brandas

[13, 14]

or at those of Hendekovic

_[15,

16].

1. Fundamental _concepts.

In this

section,

a brief summary of useful mathematical concepts relative to Fourier transform and _symmetry

_properties

is

_given.

1.1 _p-SPACE REPRESENTATION. - The

_position

space

(or

r-space)

and momentum _space

_(or

p-space)

wavefunctions are

_directly

related

_through

a Fourier transform. An orbital in

momentum is obtained from the

_position

_space

_{orbital gi (r)}

_by

the Fourier transform :

which is

_independent

of the

_spin

functions. It is

_important

to note that this transformation

preserves directions so that it is still correct in momentum _space to talk about directions

perpendicular

to the

_plane

of a

_planar

molecule as well as reflection in a _symmetry

_plane

or

rotation about a _symmetry axis of a molecule and of directions

_parallel

to certain bonds. Some

differences, however,

can be noted : the vector

_joining

two

_points

in the

_position

_space has a

direction,

a

length

and an

_origin

while in the momentum space its direction is

preserved

but information about the

origin (which corresponds

to an electron with no net

_motion)

is

meaningless.

According

to the inverse

weighting

of the Fourier transform

relation,

the electron

momentum distributions turn out to be more sensitive in the low momentum

_{(large r) region,}

and this is

_why

we can

_{loosely speak}

of an inverse

_spatial

reversal

_principle

between p- and

r-spaces

(for

a

comprehensive

review

including early

references see Ref.

[8]).

1.2 SYMMETRY PRESERVATION. - In this

section,

some relations between

symmetry

in

position

and momentum _spaces are examined :

2793

If the

_{operator S changes gl (r )}

into

_{ip (r’ ) :}

where the

_components

of the r’ vector are of the form :

with

_{( S-1}

_I)kj,

the Fourier transform

_{of 0 (r)}

is

_{given by :}

The volume element dr’ is

equal

to dr since S is a

unitary

_operator,

and the scalar

product

p . r’ can be

_expressed

as follows :

where S is the

_transposed

matrix

of S- I ;

therefore

[~ (S-1

r )]T(p )

is

_given

_{by :}

which can be

_{written 0}

_{(Sp ).}

Consequently,

if

an

_{orbital qi (r)}

has a certain _{symmetry, say}

«A »,

under

S,

an orbital

cp (p)

of symmetry o

A » under S will

_correspond

to it in the momentum _space.

This is in fact a _consequence of

_equation

_{(I) gl(r)}

_{and 0 (p)}

are

_only

different

representations

of the same vector in the _{Hilbert space ;}

_they

_possess the same

_eigenvalues

for all _operators and in

_particular

for the

_{symmetry operators}

of the molecular

_point

_group G.

Thus, 0 (r ) and .0 (p )

belong

to the same irreducible

_{representation.}

Another consequence of this

relationship

is that any nodal surface due to _symmetry is

present in both the r- and _p-space

_{representations}

of the wavefunction and

_density

functions.

This is

_obviously

related to the invariance of the

_{angular dependence}

of the molecular orbitals

(often

determined

by

spherical

harmonics)

under Fourier transform.

2.

_Symmetry

considerations in momentum _space.

2.1 ADDITIONAL INVERSION CENTER IN MOMENTUM DISTRIBUTIONS. - Let us now examine

the

_symmetry

_relationship

between the r- and _p-space electronic distributions.

An electronic distribution is

_simply

the _square modulus of the

wavefunction,

and is written

is the

_{complex conjugate of 0 (p). Although}

no

_simple

relation exists between

these two

_quantities

_and,

_{consequently,}

no immediate

_{correspondence}

between

regions

of the two distributions in both _spaces, some _symmetry relations remain. From

_{equation (7b),}

it can also be written as :

One can

write 0*(p)

in terms

of 0(-p):

If gl(r)

is

_real,

_{then ~ * (p )}

is

equal

if is _pure

_imaginary,

then

0

*

(p )

is

_equal

to - -0

(- p ).

Then

_{equation (9)}

can also be written :

Momentum densities have identical values for an electron with momentum _p and for an

electron with momentum - _p

_(i.e.

in the

_{opposite direction).}

Hence, the momentum distribution exhibits an inversion center.

_If

a molecule has a G symmetry, the

transformation

to momentum _space introduces an inversion center i in the electronic distributions

_leading

to a

(G x i )

_symmetry.

This inversion _symmetry is _necessary to ensure that the

_position

_space function

corresponding

to a

_given

momentum

_{function O(p)}

has no net

translational

motion

(otherwise

the molecule would fall

_{apart) [7].}

2.2

CONSEQUENCES

OF THE ADDITIONAL INVERSION CENTER ON MOMENTUM WAVEFUNC-TIONS. - The inversion center in the momentum

point

group of the momentum distributions entails additional

_symmetry

_properties

for the momentum wavefunctions.

Let us consider a molecule

belonging

to the

_point

_group

G,

i.e. a set of

_position

_space orbitals of the

molecule 0

(r )

transform

_according

to a certain irreducible

_{representation}

F of the _group G. If we concentrate on the case when G does not contain the inversion i, and

separate 0

into one even and one odd _component:

such that:

and :

This means that the set of

_counterparts

of

_{l/J + (r)}

in momentum _space,

_{{4> +}

_(p)}

transforms

according

to

_1’g

of the G x i and the set

_{~0- (p)}}

_according

to

_Tu

of G x i. The

_complete

momentum _space

_{orbitals, however,}

are mixtures of these _symmetry

_adapted

functions.

Furthermore,

both

_position

and momentum _space orbitals in the most

_general

case have a

2795

the real and

_imaginary

_parts of the

_{corresponding}

orbital in momentum space obtained from

equation (1)

are :

If the

_{orbital 4f (r)}

is real in the

_position

_space, i.e. =

0,

_we have from

equations

(15) :

1

So both the real and

_imaginary

_parts

of 0

(p)

are

_{eigenfunctions}

of the inversion _operator. If

the

_{wavefunction Y(r)}

is real and

symmetric

according

to

_equations

_(16),

one _gets

1m cp (p)

= 0 and the _momentum wavefunction is real and

symmetric. If ql (r)

is real and

antisymmetric 0(p)

is pure

imaginary

and

_{antisymmetric.}

Similarly,

if 4/(r)

is

_{purely imaginary,}

both the real and

_imaginary

_parts

_{of 0(p)}

are

eigenfunctions

of the inversion

_operator.

In the

_general

case, the real and

imaginary

parts of

₀

_{(p )}

can be

_split

into two components

(one

symmetric

and one

antisymmetric).

We find :

and

_{analogous expressions}

for

_Im,

and

Im_.

Putting

these

_quantities

into the

expressions

of

_{p (p ), equation (8),}

and

p (- p), equation

(11),

the

equality

between

p (p )

and

_{p (- p ), yields :}

Because the four

_components

defined in

_equations

₍₁₇₎

have

symmetry

properties

and all these four

_components

are different from _zero, for

_{equation (11)}

be

verified,

the

_following

equalities

must be true :

For these

_equalities

to be

_verified,

either

_{Re, .0 (p)}

or

_(p)

should be zero and either

Im+ 4> (p)

or should be zero

_(which

is

_denied).

Therefore both the real and

If a

molecule has a G symmetry, the

_{transformation}

to momentum _space introduces an

inversion center i in both the real and

_imaginary

_components

_of

the

_{wavefunction,}

4

(p),

but it does entail an inversion center in the

_{wavefunction ~ (p ) only}

and

_{only if}

its

_position

counterpart,

4/(r), (Eq. (14)),

is real or _pure

_imaginary.

3.

_Symmetry

correlation between

_position

and momentum _spaces.

As

_previously

mentioned,

the

_study

of the _symmetry correlation between r- and _p-spaces is

useful for

_{understanding}

the

_chemistry

_underlying

_{(e, 2e)}

_spectroscopy

for

_example.

(e, 2e)

spectroscopy

is a method for

_measuring

atomic and molecular electronic structures :

it is a coincidence

technique

based on a reaction in which an incident electron

(of

known

energy and

momentum)

knocks out an electron from a

_target

and is scattered after the collision.

_By

measuring simultaneously

the

energies

and momenta of the knocked-out electron and scattered electron after the

_collision,

and

_repeating

the

_procedure

over and over on a

_sample

of identical atoms or

_molecules,

one can obtain the momentum distribution of the knocked-out electrons before the collision. It can be shown that in the case of

symmetric

non-coplanar

(e,

2e)

experiments

the measured momentum distribution

_(i.e.

the

_(e,

_2e)

cross

section)

is

_{essentially proportional}

to the _square modulus of the momentum space molecular

orbital wave

_{function ~ (p)}

- for

an extensive

compilation

see

_[17].

Thus

_{(e, 2e)}

_{spectroscopy,}

providing

energy and

symmetry

information is very useful for

understanding

electron distributions. Since electron momentum

_{spectroscopy spectra}

can be determined as a function of momentum

_densities,

it can be used to

_diagnose

orbital

symmetry

directly.

Consequently

the theoretical

knowledge

of electronic

density

group

symmetry

in the

momentum _space is

important.

3.1 SUMMARY OF MOMENTUM SPACE AMPLITUDES CHANGES. - In view of

_exploring

and

understanding

further the

relationship

between usual

_position

_space densities and their

momentum _{space counterparts,} one has to

_keep

in mind the

_following

p-space

principles.

a)

Preservation

of

symmetry. - Orbital

symmetry is

_preserved

when

_going

from r-space to

p-space but momentum electron distributions p

gain

an additional inversion center.

b)

Inverse

_spatial

reversal. - A contraction of

probability

density

in one _space will lead to an

expansion

in the other _space and vice versa. However one has to realize that it does not

immediately imply

that

_{charge density}

in

_{large r}

_regions

_appears at

_{low p}

_regions

of

momentum

_density

and vice versa. Due to the nature of the Coulomb

_potential,

the electron

momentum is very

large

in the

_vicinity

of the nuclei and hence will appear at _very

_{large p}

in the momentum

_{density (recall}

also that the average momentum value is the

expectation

value of the

_gradient

_operator in the

_{position space),}

but one must not make the mistake of

trying

to

associate

_density

at a

_specific

_point

_p in _p-space

_directly

with

_density

at

_point

r in r-space with

I and

I p I

inversely

proportional.

c)

Molecular

_density

reversal. - The formation of

a chemical bond

_corresponds

to an _r-space

bonding density

in the direction of the bond whereas it

_corresponds

to an enhanced

density

perpendicular

to the bond in p-space.

Furthermore,

the chemical bond

_generates

a _greater

longitudinal density

at low momentum and a _greater transverse

_density

at

_high

momentum

than in the case of

_separated

atoms. First stated

_by

Coulson

_[4]

and

_{McWeeny [5],}

this

principle

has been

widely

commented - see

[8].

d)

Molecular

_density

oscillations. -

They

are the p-space manifestation of nuclear geometry

and atomic

_bonding.

A cos

_{{p ~ R /2)}

term in a minimal LCAO

_expansion

of a

_bonding

2797

planes (corresponding

to cos

_(p.

_{R /2 )}

=

0).

The location of the first nodes has been

recently

examined

_[6,

18,

19],

but the existence of an infinite number of nodal

_planes

due to the cos

term is still debated

_[20-25].

_Non-bonding

molecular orbitals do not contain much information about the nuclear

_positions

and do not show such modulations.

3.2 GROUP CORRELATION. - This section is devoted to the _summary of results

_previously

obtained and

examples

of

_experiments

are

_given.

In table I

_(columns

1 to

_3),

the _r-space

_point

_groups

_(which

reflect the

_spatial

_symmetry

properties

of the

_molecule)

have been subdivided into

_simple

_groups for which there is a

single

symmetry

operator, discrete axial

_point

groups for which there is a

_single

axis of symmetry whose order is

greater

than

two-fold,

cubic

_point

_groups which possess several

symmetry axes of order _greater than

two-fold,

and continuous

_point

_groups which contain an

axis of infinite order.

Momentum space

point

groups which reflect the momentum symmetry

properties

of the

p-space

density

are

_easily

derived. For those _r-space

_point

_groups which

_already

contain the

inversion _symmetry

_point

_groups are

_unchanged

_{in p-space, e.g.} when n is even

Cnh

=

Cn

_x

i,

when _n is odd

Dnd

=

Dn

_x i. For other

groups, p-space

point

groups are

obtained

_using

well-known results for

products

of _symmetry

_operations.

The

point

groups

relevant for momentum electron distributions are listed in column 5 of table

I,

and illustrative

examples

of molecules

_already

studied from this

_point

of view are

_given

in column 6.

3.3 GENERAL COMMENTS FOR UNDERSTANDING MOMENTUM SPACE MAPS. - Momentum

molecular

_{density plots}

can become a useful

_complement

in

_{chemistry ; they supplement}

the usual

_description

in the

_position

space. Some

rough

guidelines

are

_given

to

_help

in

interpreting

the

_experimental

_patterns. It is out of the purpose of this paper to enter the

experimental

field ;

the reader interested in the details of the

interpretation

of

_spectra

can

read to references

given.

Because the

_position

_space and momentum _space functions

belong

to the same irreducible

representation,

similarities between the two functions are to be

_expected.

But the additional inversion center in electronic momentum distributions smoothes out

_information,

which has

to be overcome

_by

the

_interpreter.

a)

As a _consequence of

_principle

a, atoms have identical r- and _p-space

_point

_groups and s-,

p- and

d-type

atomic orbitals have the same _{appearance in} both

_{representations.}

Of course,

they

do not have the same

_analytical

_behavior,

for instance a Is orbital

_roughly

behaves as

exp ( - r )

in r-space, while in p-space, it behaves as

p - 4.

The

qualitative

similarity

in both

representations

is modulated

by

the inverse

spatial

reversal

principle :

a

_position

_density

expansion

is associated with a

_{corresponding}

contraction in momentum _space

_[56].

The similar appearance of atomic orbitals has

_already

been

_pointed

out

_by

Cook et al.

_[45,

56]

who show theoretical orbitals

density

maps of atomic carbon and oxygen atoms or

_by

Leung et

al.

_[57]

who treat the case of noble gases.

The values of an atomic wavefunction at the

_origin

of _{either space in} non-zero if and

_only

if the functions

_belong

to the

_{totally symmetric representation.}

b)

The additional inversion center at the p-space

origin

changes

the overall _symmetry of the orbital from G in the

_position

_space to G x i in the momentum _space,

_leading

to

_ambiguities

in some cases.

_{Ethylene (D2h}

in

_r-space)

and

_{formaldehyde (C2v}

in

_r-space)

have both

_D2h

symmetry in the p-space

[10] ;

carbonyle

sulfide in

_r-space)

and carbon dioxide or

carbon disulfide

_(Dooh

in

_r-space)

have both

_Dcch

_symmetry in the p-space

[45].

Confusion can be removed

_by

a more detailed

analysis

of the

_experimental

results. For

Table I. - Position and momentum

2799

oxide

_[43]

momentum distributions

_by

_noticing

that nodal surfaces remain

_unchanged

in both

spaces while the additional inverse center at the p-space

origin

creates a reflection

plane

).

Conclusion.

With a view to a better

_{understanding}

of the chemical

_bonding

in the momentum _{space, the}

correlation between _symmetry

_properties

in the

_position

and momentum _{spaces has been}

studied for the wavefunctions and densities of a

_given

atom or molecule. Electronic momentum densities

_gain

an additional inversion center, whereas

only

the real and

imaginary

parts of the momentum wavefunctions exhibit inversion center

_{independently.}

Furthermore,

some

_(e,

2e) experimental

results are

_given

to show that within the above group theoretical

framework,

one can

_identify

the irreducible

_{representation}

of the

_point

group to which the wave function

_(within

the framework of the

_independent

electron model

for the

_description

of the

_molecule)

of the

_ejected

electron

_belongs.

Acknowledgments.

We thank Prof. J. L. Calais

_(University

of

_Upsala)

for

_carefully

_reading

the

_manuscript

and

giving helpful suggestions.

References

[1] EPSTEIN I. _R., Electron momentum distributions in _atoms, molecules and _{solids, Physical}

Chemistry, 2 vol. 1, Eds. A. D.

_Buckingham

and C. A. Coulson _{(Butterworths, London)} 1975.

[2] BRION C. _E., Int. J. _Quantum Chem. 29 ₍₁₉₈₆₎ 1397.

[3]

Momentum _{Distributions,} Eds. R. N. _Silver, P. E. Sokol

_(Plenum

_Press, New _York) 1989.

[4]

Cou LsoN C. A., Proc. Cambr. Philos. Soc. 37 ₍₁₉₄₁₎ 55 and 74.

[5]

MCWEENY R., Proc.

_Phys.

Soc. Lond. A 62 ₍₁₉₄₉₎ 509 and 519.

[6]

KAIJSER P., SMITH V. H. Jr., Mol. _Phys. 31 (1976) 1557.

[7]

KAIJSER P., SMITH V. H. Jr., On inversion

_Symmetry

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