Generalization of the continuous symmetry measure: the symmetry of vectors, matrices, operators and...

Generalization of the continuous symmetry measure: the symmetry

of vectors, matrices, operators and functions

Chaim Dryzun and David Avnir*

Received 8th June 2009, Accepted 23rd July 2009

First published as an Advance Article on the web 19th August 2009

DOI: 10.1039/b911179d

In this paper we generalize a method for evaluating the continuous symmetry measure, which is a

quantitative estimate of the degree of symmetry of a given object. The generalization makes it

possible to calculate the degree of symmetry content for any mathematical entity that is part of

metric spaces such as vectors, matrices, operators and functions. Furthermore, by this new

approach one can calculate the symmetry-content values for any compact symmetry groups either

finite or infinite. An advantage of the new methodology is the ability to investigate analytically

problems of symmetry changes. Examples of symmetry evaluation calculations are provided,

including mixing of ideal gases, evaluation of the symmetry content of a Hamiltonian operator,

the 2pz orbital of the hydrogen atom, and more.

1. Introduction

Symmetry plays a central role in the natural sciences, in the

arts and many other aspects of human intellectual activity. As

it links different parts of a given system it is, therefore, a

descriptor of order.1 Using symmetry one can derive easily

some central conservation laws and selection rules in physics

and chemistry.2 Exact symmetries are, however, the exception,

and structures which deviate at least to some degree from

near-symmetry abound. Since the majority of properties which

depend on exact symmetry do not shut down to zero upon

deviation from perfect symmetry, measuring the degree of

symmetry in such objects becomes interesting: it can help in

identifying quantitative correlations between property and

symmetry. Indeed, the recognition that measuring symmetry

may be useful was expressed by many authors, who also

provided various propositions as to how to carry it out.3

None, however, has been general enough to include all of

the important presentations of objects: vectors, matrices,

operators, functions. In this report we provide the most

general solution to the problem of symmetry measurement,

which encompasses all of these mathematical descriptors, and

which is analytical. Since chirality is the absence of improper

symmetries, our approach provides also the most general

solution to the problem of chirality measurement. Our general

solution addresses not only finite symmetry groups4 but also

compact (infinite) continuous groups4 (such as DNh or SU(2)).

Last but not least, since the Hamiltonian operator, H, is so

central in the natural sciences, we provide in this report a

preliminary example of how to evaluate its symmetry content.

Our starting point is the continuous symmetry measure

(CSM),5–7 which is based on a most elementary approach: it

evaluates the square of the geometrical distance between the

normalized original structure and the nearest structure which

has the desired perfect symmetry. The CSM provides a unified

scale on which different symmetries of different objects can be

compared. Since the inception of the measure5–7 it proved in

numerous publications to provide quantitative correlations

between symmetry and properties that depend on this structural

descriptor.6,8 The majority of research has been carried out on

molecular structures described through the coordinate-vectors

of their atoms. Extensions of the CSM to electron densities and

to molecular orbitals have been proposed;9,10 yet despite their

importance as pioneering efforts, they are not as general as one

may wish, and they do not adhere to the strict definition of the

CSM (as explained below in the relevant sections).

2. General methodology for calculating the

continuous symmetry measures (CSM)

2.1 The general form of the CSM

We consider a general object |Ci (in what follows, we use the

Bra-Ket notation,11 (Appendix 1) in which any structure can

be regarded as a point in a given vector space, V). To simplify

matters we assume that V is a metric space (for example

Hilbert or Banach spaces); this assures us that we can define

an inner product and a distance. We wish to calculate the

distance of this object from possessing specific G point-group

symmetry. Since size changes do not affect symmetry, we shall

use the normalized object jC0i ¼ jCiffiffiffiffiffiffiffiffiffiffihCjCip , thus ensuring that the

resulting distance values are all on the same scale. To evaluate

the G-symmetry content we define an operator, G, the

symmetry-group operator, which applies to all of the symmetry

operations of G on the studied object, then averages the results,

returning a new object, |Oi, in the same vector space G: V - V

(a more explicit, mathematical definition of the symmetry-

group operator, G, is presented later in sections 2.4 and 2.5):

jOi ¼ GjC0i ¼ GjCiffiffiffiffiffiffiffiffiffiffiffiffiffihCjCi

p ð1ÞInstitute of Chemistry and The Lise Meitner Minerva Center forComputational Quantum Chemistry, The Hebrew University ofJerusalem, Jerusalem 91904, Israel. E-mail: [email protected]

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 | 9653

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics

The original object, |Ci, is G-symmetric if the symmetry-group

operator, G, leaves it unchanged (that is, if |C0i is an eigen-

vector of G with eigenvalue 1, |C0i = |Oi). If a change does

occur upon applying G then the magnitude of that change is a

measure of the degree of G-symmetry in |Ci.Each of the symmetry operations of G, namely gi, can be

represented by a symmetry-operation operator, gi, in the same

vector space as the original object, gi: V - V. The symmetry-

operation operators are unitary and preserve the size and the

shape of the original object. If the operation is a proper

symmetry operation, it can change only the spatial orientation

of the object. If the operation is an improper symmetry

operation, it changes the spatial orientation of the object

and then multiplies it by �1, causing an internal flip. The

symmetry operations divide the vector space in a G-symmetrical

way. For cyclic groups this is clear, because all of the symmetry

operations are different powers of one basic operation, and

thus one applies the same basic operation several times,

retaining the magnitude and direction of the operation each

time. After applying the basic operation h times (h is the order,

or the size, of the group), the original object is reached again.

Symmetry groups that are not cyclic can be constructed by

multiplying several cyclic groups, so the above is true for each

of the cyclic sub-groups, and therefore for the whole G group.

To summarize, by applying all of the symmetry operations of

the G symmetry group on the original normalized structure,

we get a collection of h resulting objects, |oii = gi|C0i, andthat collection has the desired G symmetry. We can now take

the collection and average it, getting one new object, |Oi,which must have the desired G symmetry. An equivalent

approach is to take the collection of h resulting objects,

calculate the distances between them, and then average the

result. It has been previously shown7 that the two approaches

are indeed equivalent.

Next we express the change imposed on the original structure

thus leading to |Oi in terms of the expectation value of the

symmetry group operator, hGi, namely, we estimate the level

of (normalized) overlap between the original structure and the

symmetric structure. We get a value that ranges between zero

(no overlap) to 1 (complete overlap):

0 � hGi ¼ hC0jGjC0i ¼ hCjGjCihCjCi ¼ hC0jOi � 1:

This expectation value, hGi, is a similarity measure between

the original object and the closest (see section 2.2) structure

under all symmetry transformations of G (we call it, in short,

the closest G-symmetrical object), and therefore is a symmetry

measure (note that the orientation of G must be optimized).

Another way to interpret hGi is as the estimation of the level of

overlap between the original structure taken h times, and all

the resulting objects, |oii, after normalizing to size and to the

number of symmetry operations. This overlap value expresses

the average similarity between the original object and all

the resulting objects, |oii; again, the two approaches are

equivalent.

Whereas hGi is a symmetry measure in the full sense of the

word and which is based on similarity, the CSM was defined as

the distance from the desired symmetry, and therefore it is a

dissimilarity measure. In order to comply with all previous

literature, we shall use therefore the following expression:

SðGÞ ¼ 1� hGi ¼ 1� hCjGjCihCjCi : ð2Þ

Here, a zero value for S(G) means that the object is of exact

G symmetry. (For convenience and for compliance with earlier

reports,5–8 the range between zero to one has been expanded

by a factor of 100; however we shall omit this factor in the

following.)

Finally, since by our method one has to be able to operate

all of the symmetry operations of the symmetry group, it can

be used for finite symmetry groups or for compact continuous

symmetry groups,4 but not for general infinite groups4 (such as

translation symmetry, space groups and more).

2.2 Eqn (2) is a CSM equation

Next we show that that S(G) fulfils the general CSM definition,

which requires that |Oi be the closest object to |C0i under all ofthe symmetry transformations of G; and that eqn (2) is

equivalent to the original CSM equation (eqn (3)).

Originally5–7 S(G) was defined as the square of the distances

between those objects:

SðGÞ ¼ NXNk¼1jQk � Pkj2: ð3aÞ

Here Qk are the coordinate vectors of the original structure

(|Ci, in our general terminology), Pk are the coordinate

vectors of the closest G-symmetrical structure, and N is the

size normalization factor of the original structure:

N ¼ 1

jQj2¼ 1PN

i¼1 Q2i

:

We now show that |Oi is indeed the closest object to |C0inamely, that it is Pk of the definition of eqn (3a). |Cirepresents the coefficients of Qk in the 3D Euclidian space

and the normalization factor becomes N ¼ 1hCjCi. Let us multiply

|Oi by a scaling factor, a, and assume that a|Oi is the closest

object to |C0i, so it represents the coefficients of Pk. Eqn (3a)

becomes:

S(G) = N||Ci � aG|Ci|2

= N(hC|Ci � ahC|G|Ci�ahC|G*|Ci+a2hC|G*G|Ci)(3b)

where G* is the conjugated transpose of G. The symmetry

group operator, G, is Hermitian (because it is built as a sum

of unitary square operators, and G has a group structure), so

G= G*. We know that |Oi = G|Ci and because, as already

showed, |Oi has the desired G symmetry, then |Oi = G|Oi.This means that G2 = GG = G, so the symmetry group

operator, G, is a projection (and because it is also Hermitian it

is an orthogonal projection). Eqn (3b) becomes:

SðGÞ ¼ NðhCjCi þ ða2 � 2aÞhCjGjCiÞ

¼ 1þ ða2 � 2aÞ hCjGjCihCjCi :ð3cÞ

9654 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009

Minimizing S(G) with respect to a:

0 ¼ @SðGÞ@a

¼ ð2a� 2Þ hCjGjCihCjCi2a = 2 ) a = 1

In a similar way one can show that the only unitary

transformation that minimizes S(G) is the identity transformation,

I. This proves that eqn (3a) and (3b) fulfil the general CSM

definition, namely that |Oi be the closest to |C0i under all ofthe symmetry transformations of G. (We will not consider

non-unitary transformations because they distort the object

and we can not regard the resulting object as containing all the

symmetry transformations of the given symmetry group, G.)

Next we show the equivalence of eqn (2) and (3a): From the

above we see that hC|GG|Ci= hC|G|Ci. Introducing this intoeqn (3c) we get:

S(G) = N(hC|Ci � hC|G|Ci)

where N ¼ 1hCjCi, and therefore SðGÞ ¼ 1� hCjGjCihCjCi .

Since the expression hCjGjCihCjCi is the expectation value of the

symmetry group operator: hCjGjCihCjCi ¼ G we get back eqn (2).

2.3 Graphical representation of CSM

Having shown that the distance between |C0i and |Oi is minimal,

the relation between the various variables can be presented

graphically as a right-angled triangle (Fig. 1): at the origin we

place the center of mass of the object. Then, we have two vectors,

one is |C0i the size of which is 1, and the other is |Oi the size of

which isffiffiffiffiffiffiffiffiffiffiffiffihOjOi

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihC0jGGjC0i

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihC0jGjC0i

q¼

ffiffiffiffiffiffiffiffihGi

q.

Therefore the third side of the triangle has a size of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� hGi

q(Fig. 1). The CSM is the square of the distance between |C0i and|Oi, which is 1 � hGi (eqn (2)).

2.4 The symmetry measure for finite groups

For finite symmetry groups the explicit expression of eqn (3)

takes into account the h symmetry operations of the symmetry

group G: g1. . .gh, for which, as described above, each gi

can be represented by a symmetry operator gi, i = 1,2. . .h:

G ¼ 1h

Pi

gi, and eqn (2) becomes:

SðGÞ ¼ 1� hGi ¼ 1�

Phi¼1hCjgijCi

hhCjCi ð4Þ

2.5 The symmetry measure for infinite groups—final

generalization

A more general situation is of a compact continuous infinite

topological symmetry group4 where the symmetry operations

are associated with the k independent symmetry elements

(or, equivalently, as symmetry cyclic sub-groups). Each of

the k sub-groups contains an infinite number of symmetry

operations that can be represented as a single operation, g(ai),

which depends on a continuous parameter, ai, which is

bounded in the range Ai r ai r Bi. The dimension (‘‘group

order’’) of the ith sub-group is the integral over the range of

the parameter:RBi

Ai

dai. The dimension of the symmetry group,

G, is the product of the dimensions of all the sub-groups:Qki¼1

RBi

Ai

dai. To obtain all of the symmetry operations of G from

the operations of the sub-groups one must multiply all of the

symmetry operations of all sub-groups. The symmetry group

operator in this case is

G ¼

Qki¼1

RBiAi

gðaiÞdai

Qki¼1

RBiAi

dai

;

and thus eqn (2) becomes:

SðGÞ ¼ 1� hGi ¼ 1�

Qki¼1

RBi

Ai

hCjgðaiÞjCi dai

hCjCiQki¼1

RBi

Ai

dai

: ð5Þ

If the sub-groups are finite, the integrations in eqn (5) indicate

summations over all of the sub-groups’ symmetry operators.

Since the product of several summations is a new summation,

if all the sub-groups are finite, eqn (5) becomes eqn (4).

Therefore eqn (5) is the most general expression of the

CSM: it can be used to solve purely finite groups (such as

Cn, Cs, Dnh), purely infinite compact groups (such as SO(3))

and mixed groups (such as CNv, DNh).

2.6 Some practical aspects of the calculation

* The measure is invariant under unitary transformation: If |Ciis the original object and U is a unitary operator then

|C1i= U|Ci is the transformed object. When the space group

operator, G, operates on the original object, |Ci, the

expectation value is hGi ¼ hCjGjCihCjCi . When the space group

operator, G, operates on the transformed object, |C1i = U|Ci,the expectation value is:

hG1i ¼hC1jGjC1ihC1jC1i

¼ hCjU�GUjCi

hCjU�UjCi

¼ hCjU�UGjCi

hCjCi ¼ hCjGjCihCjCi ¼ hGi:Fig. 1 Graphical representation of the general method for calculating

CSM.


This means that the expectation value of the space group

operator, G, is invariant under unitary transformation, and

therefore also the symmetry measure, S(G) = 1 � hGi, is

invariant under unitary transformation.

* The origin of the system: The symmetry elements of the

symmetry group must pass through the center of mass of the

object |Ci, but it is more convenient to use the origin of the

space. Therefore we move the center of mass of the object to

the origin before performing the symmetry calculation.

* The identity operation: It should be noticed that in

practice, not all of the symmetry operators have to be actually

operated when calculating hGi ¼ hCjGjCihhCjCi . When the identity

operation, E, is applied the result is, of course, the same object:

|Ci = E|Ci. In this case, we know that always: hC|E|Ci =hC|Ci and therefore hCjEjCi

hhCjCi ¼ 1h.

* The inverse symmetry-operation operator: For each

symmetry-operation operator, gi, within G there must exist

its inverse operator, gj, (i and j can be equal), such that

gigj = gjgi = E. It was previously shown12 that if gi and gjare inverse to each other, then their expectation values are

equal: hC|gi|Ci = hC|gj|Ci. So if gi a gj but they are still

inverse to each other it is enough to calculate only the

expressions for one of the two symmetry operations, hCjgi jCihhCjCi

and multiply it by 2.

* Permutations: For calculating the symmetry measure of an

object that is constructed by some assembly (number of

vectors, number of functions), we have to find the best

permutation, that is, the best pairing between the original

assembly’s members to the assembly’s members after a

symmetry operation had been operated. It should be noticed

that not all permutations are allowed—only the symmetry

preserving permutations should be considered. For a detailed

discussion of the permutation issue, see ref. 7.

3. Symmetry measures of vectors

3.1 Structures represented by one vector

After formulating a general expression for CSM, we will now show

how one can apply it to various mathematical entities, starting with

structures that can be described by one vector. If the object is a

single vector, �Q, then for finite groups eqn (4) becomes:

SðGÞ ¼ 1�

Phi¼1

�Q�gi �Q

hj �Qj2ð6aÞ

where �Q* is the conjugated transpose of the vector �Q. For the

general case of infinite compact groups, eqn (5) becomes:

SðGÞ ¼ 1�

Qki¼1

RBi

Ai

�Q�gðaiÞ �Q dai

j �Qj2Qki¼1

RBi

Ai

dai

ð6bÞ

3.2 Structures represented by assembly of coordinate vectors

We recall that a vectorial description of molecules has

been the (only) application we developed in previous

publications.5–8 We describe a molecule containing N-atoms

as an assembly of N 3D coordinate-vectors (containing real

numbers only),

�Qj ¼xjyjzj

0@

1A;

j = 1. . .N. For calculating the symmetry of such an assembly,

we have to find the best permutation, that is, the best pairing

of vectors between the original set of vectors and the vectors

after a symmetry operation had been operated. (Such

permutation, or pairing, is necessary only when the system is

being described by an assembly rather then by only one

mathematical object.) A specific label-permuted object is then�V 0j,i = giPj,i

�Qj = Ri�Q0j,i where the matrix Pj,i determines the

permutation ( �Qj and �V 0j,i are of the same size). We are seeking

the one permutation that maximizes hGi and minimizes S(G).

For finite groups eqn (4) becomes then:

SðGÞ ¼ 1�max

Phi¼1

PNj¼1

�QTj gi

�Q0j;i

hPNj¼1j �Qj j

2

ð7aÞ

This is also the general equation that appears in our previous

report,7 which presents analytical solutions for several specific

G point groups.13

For the general case of infinite compact groups, eqn (5)

becomes:

SðGÞ ¼ 1�max

PNj¼1

Qki¼1

RBi

Ai

�QTj gðaiÞ �Q

0j;ai

dai

Qki¼1

RBi

Ai

daiPNj¼1j �Qj j

2

ð7bÞ

3.3 Center of mass

As mentioned in section 2.6, it is convenient to locate the

center of mass of the assembly at the origin. Thus, given an

assembly of vectors,

�Wj ¼ajbjcj

0@

1A;

with the center of mass located at

�Q0 ¼

x0 ¼ 1N

PNi¼1

ai

y0 ¼ 1N

PNi¼1

bi

z0 ¼ 1N

PNi¼1

ci

0BBBBBBB@

1CCCCCCCA;


translating it to the origin results in:

�Qj ¼ �Wj � �Q00 ¼

aj � x0bj � y0cj � z0

0@

1A ¼

aj � 1N

PNi¼1

ai

bj � 1N

PNi¼1

bi

cj � 1N

PNi¼1

ci

0BBBBBBB@

1CCCCCCCA:

3.4 Vectors example: AB2 type molecule

Consider an AB2 type molecule with a head angle a, as shownin Fig. 2. The 2D vectors that represent the atoms are:

A 0;2

3cos

a2

� ��

B sina2

� �;� 1

3cos

a2

� ��

B � sina2

� �;� 1

3cos

a2

� ��

For this molecule we wish to calculate S(Ci) as a function of

the head angle, a. The inversion symmetry group contains two

symmetry operations: the identity operation, E, for which

E �Q = �Q, and the inversion operation, ı, for which

ı �Q = � �Q. In our case:

i 0;2

3cos

a2

� �� ! 0;� 2

3cos

a2

� ��

i sina2

� �;� 1

3cos

a2

� �� ! � sin

a2

� �;1

3cos

a2

� ��

i � sina2

� �;� 1

3cos

a2

� �� ! sin

a2

� �;1

3cos

a2

� ��

The single A atom can permute only with itself. The two B

atoms provide two possible permutations: Each B atom

permutes with itself; and the two B atoms permute with each

other. We will examine both permutations and choose the one

that returns the minimal value. From eqn (7a):

SðC iÞ ¼ 1�

Pj

�Q�j E

�Qj þPj

�Q�j i

�Qj

2Pj

j �Qj j2

;

for which the terms are

Xj

j �Qj j2 ¼ 2

3cos2

a2

� �þ 2 sin2

a2

� �

Xj

�Q�j E

�Qj ¼Xj

j �Qj j2 ¼ 2

3cos2

a2

� �þ 2 sin2

a2

� �:

The last term provides two possibilities. If each B atom

permutes with itself, then:

Xj

�Q�j i

�Qj ¼Xj

j �Qj j2 ¼ � 2

3cos2

a2

� �� 2 sin2

a2

� �;

and the CSM value is always S(Ci) = 1, which is the maximal

value. If the two B atoms permute with each other, then:

Xj

�Q�j i

�Qj ¼Xj

j �Qj j2 ¼ � 2

3cos2

a2

� �þ 2 sin2

a2

� �;

and the CSM value is

SðC iÞ ¼cos2 a

2

� �1þ 2 sin2 a

2

� � :The dependence of S(Ci) on a is shown in Fig. 2. As expected,

it is seen that the molecule has inversion symmetry at a = 1801,

and is farthest from it when the angle closes to zero. These

extreme values can be obtained directly by differentiating the

last equation with respect to a (see Appendix 2).

4. Symmetry measures of matrices and operators

4.1 General treatment

Next we advance to the more general case of matrices and

operators (which are a different manifestation of the same

mathematical systems). The inner product of matrices

and operators is defined as Frobenius inner product:14

hAjBi ¼Pi

Pj

AijBij ¼ trðA�BÞ ¼ trðAB�Þ where A* is the

conjugate transpose of A, and trðAÞ ¼Pi

aii is the trace A.

For a given system (represented by a matrix or an operator),

M, and for finite symmetry groups eqn (4) becomes:

SðGÞ ¼ 1�

Phi¼1

trðM�giMÞ

htrðM�MÞ ; ð8aÞ

For infinite compact symmetry point groups’ eqn (5) becomes:

SðGÞ ¼ 1�

Qki¼1

RBi

Ai

trðM�gðaiÞMÞ dai

trðM�MÞQki¼1

RBi

Ai

dai

: ð8bÞFig. 2 The inversion symmetry measure, S(Ci) for AB2 type mole-

cules, as a function of the head angle a.


The symmetry operations are represented by a unitary

superoperators,15 gi. The simplest way to define the

operation of this superoperator is through the following

multiplication (of matrices or of operators): giM = giM = W.

In quantum physics superoperators are sometimes

defined by the commutation relation giM = giMgi�1 = W.16

In these cases, if gi is a superoperator representing some

symmetry operation and M = giMgi�1 = W equality

holds, it is said that M is invariant under the gi symmetry

transformation. This is another way to define the symmetry

superoperator.

There are two possible types of symmetry operations in the

context of matrices: The first type contains symmetry

operations that rearrange the structure of the matrix or the

operator. For example, a general 2 � 2 matrix is

M ¼ a bc d

� �. The inversion symmetry operation replaces

the mij entry with the m3�i,3�j entry, so the resulting matrix

is W ¼ d cb a

� �(this operation can be represented by the

matrix Ci ¼0 11 0

� �. As one can check, CiMCi

�1 = W). The

second type is symmetry operations that change the value of

the different entries of the matrix or the operator in a specific

way. For example, one can define the inversion symmetry

operation as a symmetry operation that multiplies each entry

in the matrix by (�1) factor, so the resulting matrix is

W ¼ �a �b�c �d

� �(this operation can be represented by the

matrix Ci ¼�1 00 �1

� �. As one can check, CiM = W).

As in the previous cases, the symmetry elements of the

group must pass through the center of mass of the system,

and again, it is more convenient to work with symmetry

elements that pass through the origin. Finding the center of

mass of matrices and operators is not always trivial and

there is no general way for doing this. One has to inspect the

system that is represented by the matrix or by operator

in order to determine what is the meaning of the center of

mass and how can be found it; this is exemplified in the next

section.

Finally, in relation to the previous Section on vectors, note

that indeed, one can extend the vectors treatment to matrices,

but not all matrices can be reduced to vectors, such as energy-

describing matrices.

4.2 Matrix example: mixing of ideal gases

Our example is the mixing of two ideal gases:17 A 2D container

with a barrier in the middle has on each side a different ideal

gas (denoted as 1 and 2). We remove the barrier, and the two

gases mix, driven only by entropy increase. This system can be

represented as a matrix, such that each matrix cell represents a

specific position in plane of either gas atom 1 or gas atom 2.

Mixing means that the atoms can change matrix cells. In this

case, we are dealing with the fist type of symmetry operation,

where the symmetry operation rearranges the structure of the

matrix. For simplicity let us look at a 4 � 4 matrix. There are

two extreme cases. First, there is the initial case, where each

gas is concentrated at one side of the system:

M1 ¼

1 1 2 21 1 2 21 1 2 21 1 2 2

0BB@

1CCA:

This system has Cs symmetry, where the mirror plane bisects

horizontally the matrix, and there is a unitary matrix U1 that

fulfills U1M1U1�1 = M1). The second extreme state is of the

full ‘‘crystalline’’ mixing case:

M2 ¼

1 2 1 22 1 2 11 2 1 22 1 2 1

0BB@

1CCA

which resemble the arrangement of NaCl crystal. This system

has Ci symmetry and here too there is a unitary matrix U2 that

fulfills U2M2U2�1 = M2. There are many possible inter-

mediate arrangements that, in most cases, have C1 symmetry,

for example:

M3 ¼

1 2 1 12 1 2 21 1 1 21 2 2 2

0BB@

1CCA:

In this case

U1M3U�11 ¼

1 2 2 21 1 1 22 1 2 21 2 1 1

0BB@

1CCAaM3

and

U2M3U�12 ¼

2 2 2 12 1 1 12 2 1 21 1 2 1

0BB@

1CCAaM3:

Thus, we can calculate the S(Cs) or S(Ci) for any arrangement,

using U1 or U2.

Let us now calculate S(Cs) and S(Ci) of M3: The first step is

to move the center of mass of the system to the origin. In this

case, we need to ensure that the outcome does not depend on

the type of particles, or in other words—that we can write as

matrix elements, e.g., 3 instead of 2 and the outcome will not

change. The first step is to bring the matrix to a form where the

sum of all the matrix entries is zero. To do so we first calculate

the shift parameter a:

a ¼

P4i¼1

P4j¼1

mi;j

P4i¼1

P4j¼1

1

¼ 24

16¼ 1:5:


We subtract this value from each cell of the matrix:

W3 ¼M3 � a1 ¼

�0:5 0:5 �0:5 �0:50:5 �0:5 0:5 0:5�0:5 �0:5 �0:5 0:5�0:5 0:5 0:5 0:5

0BB@

1CCA:

The normalization term, tr(M*M), ensures us that the result

will be the same, regardless of the type of particles

(for instance, we can denote one gas as 1 and the other as 3).

Next we can calculate S(Cs) and S(Ci) from:

SðCsÞ ¼ 1� trðW�3 ðIW3I

�1ÞÞ þ trðW�3 ðU1W3U

�11 ÞÞ

2trðW�3W3Þ

SðCiÞ ¼ 1� trðW�3 ðIW3I


�12 ÞÞ

2trðW�3W3Þ

It is easy to calculate the normalization factor and the identity

term tr(W*3(IW3I�1)), which for all three matricesM1,M2 and

M3, is the same:

tr(W*iWi) = tr(W*i(IWiI�1)) = 4 i = 1,2,3

Next we calculate the mirror-symmetry term,

tr(W*3(U1W3U1�1)) and the inversion-symmetry term,

tr(W*3(U2W3U2�1)):

tr(W*3(U1W3U1�1)) = 0, tr(W*3(U2W3U2

�1)) = 0

The S(Cs) and S(Ci) for the studies system are then:

SðCsÞ ¼ 1� trðW�3 ðIW3I


�11 ÞÞ

2trðW�3W3Þ

¼ 1� 4þ 0

2 � 4 ¼ 0:5



�12 ÞÞ

2trðW�3W3Þ

¼ 1� 4þ 0

2 � 4 ¼ 0:5

Finally, we define a new matrix, D = S(Ci)S(Cs). The matrix

which has the highest D value is the one that is the farthest

from both Cs and Ci symmetries and therefore represents

the least ordered arrangement. The D value of M3 is

D3 = S(Ci)S(Cs) = 0.5 � 0.5 = 0.25. Likewise one can screen

over all possible Mi matrices (for instance through S(Cs)/S(Ci)

symmetry maps18) to identify the highest D value. In Appendix 3

we show that, as expected, the D values of M1, M2 are zero.

Note the simplicity of this calculation, compared to entropy

calculation in determining disorder.

4.3 Operator example: the Hamiltonian of the hydrogen atom

The most obvious example for the application of the

symmetry measure to operators is to evaluate the symmetry

content of a Hamiltonian operator, H. We exemplify it for

the Hamiltonian of the hydrogen atom19 (using spherical

coordinates):

H0 ¼ ��h2

2mr2@

@rr2@

@r

� �þ 1

sin y@

@ysin y

@

@y

� �þ 1

sin2 y

@2

@2f

where y is the polar angle from the z-axis (0r yr p) and f is

the azimuthal angle in the xy-plane (0 r y r p). This

Hamiltonian is isotropic, so we can rotate around the z axis

by an angle of d, an operation which is represented by the

superoperator: Rz(d): f - f + d.20 It is clear that this

symmetry operation only changes the angle f, by adding a

constant to it. In the Hamiltonian only the last term depends

on f, and adding a constant to it does not change the value of

the derivative. Therefore the Hamiltonian is unchanged by the

rotation (it is invariant under this symmetry operation):

Rz(d)H0Rz�1(d) = H0. The expectation value of the

symmetry-group operator is:

hC1;zi ¼

R2p0

trðH�0ðRzðdÞH0R�1z ðdÞÞÞ dd

trðH�0H0ÞR2p0

dd

¼

R2p0

trðH�0H0Þ dd

trðH�0H0ÞR2p0

dd

¼ 1

and the symmetry measure is zero, as expected: S(CN,z) =

1 � hCN,zi = 0.

We now build a new Hamiltonian that contains also a

potential directed along the x-axis:

H ¼ H0 þ H1

H1 ¼ kx2 ¼ kr2 sin2 y cos2 f

H ¼ � �h2

2mr2@

@rr2@

@r

� �þ 1

sin y@

@ysin y

@

@y

� �

þ 1

sin2 y

@2

@2f

þ kr2 sin2 y cos2 f

ð9Þ

The S(CN,z) of this Hamiltonian is not zero. We shall

evaluate the measure of a one level system, namely only one

wave function, the 1S orbital, with quantum numbers n = 1,

l = 0, ml = 0 (we will work with atomic units):

jC1Si ¼ 1ffiffipp expð�rÞ.19 The expectation value of the symmetry

group operator is:

hC1;zi ¼

R2p0

trðH�ðRzðdÞHR�1z ðdÞÞÞ dd

trðH�HÞR2p0

dd

:


The normalization factor can be written using the following

expressions:

trðH�0H0Þ ¼ hC1SjH�0H0jC1Si ¼

E2H

4

trðH�0H0Þ ¼ hC1SjH�0H0jC1Si ¼ trðH�0H0Þ

¼ hC1SjH�0H0jC1Si ¼ �

EH

2pk

trðH�1H1Þ ¼ hC1SjH�0H0jC1Si ¼

9

2k2

We know that Rz(d)H0Rz�1(d) = H0 and

R2p0

dd ¼ 2p,therefore:

Z2p0


¼ trðH�0 � H0ÞZ2p0

dd ¼ pE2H

2

Z2p0


¼ trðH�1H0ÞZ2p0

dd ¼ �EHk

RzðdÞH1R�1z ðdÞ ¼ kr2 sin2 y½cos2 f cos2 dþ sin2 f sin2 d

� 12sinð2fÞ sinð2dÞ�

Z2p0

trðH�0ðRzðdÞH1R�1z ðdÞÞÞ dd ¼ �pEHk

Z2p0

trðH�1ðRzðdÞH1R�1z ðdÞÞÞ dd ¼ 6pk2

hC1;zi ¼pE2

H � 2ð1þ pÞEHkþ 12pk2

pE2H � 4EHkþ 18pk2

And the symmetry measure is:

SðC1;zÞ ¼ 1� hC1;zi ¼2ðp� 1ÞEHkþ 6pk2


ð10Þ

We can see that the symmetry measure depends on k.

For k = 0, S(CN,z) = 0, as expected. When k - N,

SðC1;zÞ ! 6pk2

18pk2 ¼13. This value is also logical—when

k - N the Hamiltonian is like a vector in the x direction.

We know that the distance of a perfect vector, a line, in the

x direction from an isotropic circular potential is 1/3.

Analytical investigation of S(CN,z) with respect to k shows

(Appendix 4 and Fig. 3) that the S(CN,z) measure has two

extrema: a local minimum at k = 0 (where S(CN,z) = 0); and

a maximum at k E 12.63 (where S(CN,z) = 0.45).

An integral part of the CSM analysis is to determine what

the closet object with the desired G symmetry is. This can be

done easily using eqn (1) and (5). Let us demonstrate it for the

Hamiltonian of eqn (9): what is the closest operator to this

Hamiltonian that has CN,z symmetry? We can build this

symmetric-operator as follows:

Hsym ¼

R2p0

ðRzðdÞHR�1z ðdÞÞ dd

R2p0

dd

¼

R2p0

ðRzðdÞH0R�1z ðdÞÞ ddþ

R2p0

RzðdÞðkr2 sin2 y cos2 fÞ dd

R2p0

dd

¼H0

R2p0

ddþ kr2 sin2 yR2p0

cos2ðfþ dÞ dd

R2p0

dd

Fig. 3 The degree of rotational S(CN,z) symmetry (rotation around

the z axis) of the one electron hydrogen atom Hamiltonian of eqn (9)

as a function of the parameter k (which is the magnitude of the

potential directed along the x-axis (eqn (9) and (10))).


¼ H0 þk � r2 sin2 y cos2 f

R2p0

cos2 d ddþ sin2 fR2p0

sin2 d dd� 2 sinð2fÞR2p0

sinð2dÞ dd" #

R2p0

dd

¼ H0 þpkr2 sin2 yðcos2 fþ sin2 fÞ

2p

¼ H0 þpkr2 sin2 y

2p

Hsym ¼ H0 þ1

2kr2 sin2 y ¼ H0 þ

1

2kðx2 þ y2Þ

Since this operator is isotropic in the xy plane it must be CN,z

symmetric. If no field is applied in the x direction then k = 0

and it is clear that H = Hsym = H0. In a forthcoming report

we extend this approach to discuss diatomic Huckel molecules.

5. Symmetry measures of functions

Last, but not least, we apply the general CSM formulation for

functions. If our object is a function, f(�x), then for finite

groups eqn (4) becomes:

SðGÞ ¼ 1�

Phi¼1

RD

f �ð�xÞgif ð�xÞ d�x

hRD

f �ð�xÞf ð�xÞ d�x; ð11aÞ

where f *(�x) is the conjugated function to f(�x). For infinite

compact symmetry point groups, eqn (5) becomes:

SðGÞ ¼ 1�

Qki¼1

RBi

Ai

RD

f �ð�xÞgðaiÞf ð�xÞ d�x dai

RD

f �ð�xÞf ð�xÞ d�xQki¼1

RBi

Ai

dai

: ð11bÞ

In order to be able to calculate the symmetry measures, the

function must be integrable. Most physical systems are integr-

able, so this is not a serious limitation of the method.

Note that when we take a function in the vector-representation

limit, eqn (11a) and (11b) will not behave as eqn (6a)–(7b).

This is understandable from the physics point of view: A

quantum system that is described by functions can never

become a real classical system which is described by vectors.

As in all previous cases, it is convenient to locate the center

of mass of the function at the origin. Shifting f(�y) to the origin

is done through the following procedure: First we find the

location of the center of mass of the function:

yj;0 ¼ hyji ¼ hf jyj jf i ¼RDj

f �ð�yÞyjf ð�yÞ dyj. Now we can define

a new variable set, �x, through xj = yj � yj,0, thus locating the

center of mass of f(�x) at the origin. One must pay attention

that the integration range of the new variable set, �x, is similarly

shifted with respect to the integration range of �y.

Because of the central role of wave functions in the

quantum-mechanical description of nature, we treat them

separately in the next section.

6. Symmetry measure of wave-functions

6.1 General comments

The symmetry of wave-function is one of their key

characteristics; the CSM approach provides a new way to

investigate electronic phenomena from this point of view. In

most cases, the analytical solutions to wave-functions are

unavailable; however, when the Hamiltonian can be solved

analytically, we can apply eqn (11a) or (11b) directly. In

other cases, several options exist as for the description of

wave-functions which can be used for CSM calculations:

* Using numerical matrix description of the wave-function:

Here the wave-function is represented by a 3D grid. The value

at each grid point represents the value of the wave-function in

a certain point in space, in the limits of the data. We are using

only the main part of the wave function, where it has

significant value. A common practice is to use Gaussian’s

cube files.21 In this case the method described in section 5 can

be used directly on the matrix.

* Using valence bond (VB) description of wave-functions: In

VB22 the wave-function is composed as a linear combination

of some basic functions with different coefficients. These VB

functions represent different weights of ionic and covalent

structures of the molecule. Thus, the wave-function can be

written as:

hxjCi ¼ f ð�xÞ ¼Xk

Ck � Fkð�xÞ:

The wave-function is always normalized:

hCjCi ¼ZD

f �ð�xÞf ð�xÞ d�x ¼ 1:

It follows from eqn (11a) that the CSM for finite point

groups is:

hCjgijCi ¼ZD

f �ð�xÞgi f ð�xÞ d�x

¼Xk

Xl

CkCl

ZD

F�l ð�xÞgiFkð�xÞ d�x

SðGÞ ¼ 1� 1

h

Xhi¼1

Xk

Xl

CkCl

ZD

F�l ð�xÞgiFkð�xÞ d�x

ð12aÞ


And for infinite compact groups the symmetry measure is:

hCjgðaiÞjCi ¼ZD

f �ð�xÞgi f ð�xÞ d�x

¼Xk

Xl

CkCl

ZD

F�l ð�xÞgðaiÞFkð�xÞ d�x

SðGÞ ¼1�

Pk

Pl

CkCl

Qki¼1

RBi

Ai

RD

F�l ð�xÞgðaiÞFkð�xÞ d�x dai

Qki¼1

RBi

Ai

dai

ð12bÞ

* Using the Kohn–Sham orbitals in density functional theory

(DFT): We recall that in DFT, one uses the Kohn–Sham

orbitals23 that are mathematically built in a similar way to

molecular orbitals (MO)s.24 where the energy, E[r], is

expressed in terms of the electron density, r(�r), which, in turn,

is composed of a sum of the densities of the isolated electrons:

rð�rÞ ¼Pz

jCzj2. In order to get an initial density (that can later

be improved) we assume that the densities of the isolated

electrons, |Cz|2, can be expressed as one-electron orbitals in

Slater determinants of an effective operator. In other words,

|Cz|2 can be composed as antisymmetrized product of a basis

set, designated as Fr(�x), each of which is occupied by Zrelectrons. These one-electron orbitals comprise an orthogonal set:Z

D

F�r ð�xÞFsð�xÞ d�x ¼ dr;s:

Each one-electron orbital is expressed as linear combination

of basis-set functions:

Frð�xÞ ¼Xk

Cr;k fkð�xÞ:

Our main treatment, next, is of MO functions.24 As the basic

mathematical structure of DFT and MO is similar (although

the functions are different and have different meanings) we can

use eqn (13a) and (13b) of the following section, replacing the

MO functions with DFT functions.

6.2 Using molecular orbitals (MO) description of

wave-functions

Here we follow the basic idea proposed by Grimme9 and

extend it. We recall that according to the MO approach24

the wave-function is composed from an antisymmetrized

multiplication of MO functions, which form a basic. The rth

MO, Fr(�x), is occupied by Zr electrons (in Hartree–Fock, HF,

methods Zr is integer;9,24 in CI methods9,24 Zr can be

non-integer). The MOs comprise an orthogonal set, namelyRD

F�r ð�xÞFsð�xÞ d�x ¼ dr;s. Each MO is composed of a linear

combination of basis-set functions, the atomic orbitals

(AOs): Frð�xÞ ¼Pk

Cr;kfkð�xÞ. Instead of using the exact Slater

determinants, which is more precise but much more difficult,

we use the standard way of treating the orbitals, following

Grimme.9

Since the symmetry-operation operators (gi or g(ai)) are

unitary, applying them on an MO results in a new function

that is not part of the orthogonal MO basis set; it will be a

linear combination of all the MOs. Therefore, it is important

to compare the resulting function with all the MOs.

In order to evaluate the CSM of a wave function we apply

the symmetry-operations operator on the MO and multiply by

its occupation number, overlapping the result with all MOs.

This is repeated for all MOs and the results are summed, from

which, for finite symmetry point groups, we obtain (recall that

the wave-function is normalized: hCjCi ¼RD

f �ð�xÞf ð�xÞ d�x ¼ 1):

hCjgijCi Xr

Xs

ZrhFsjgijFri

¼Xr

Xs

Zr

ZD

F�s ð�xÞgiFrð�xÞ d�x

¼Xr

Xs

Xk

Xl

ZrCr;kCs;l

ZD

f�l ð�xÞgifkð�xÞ d�x

hCjGjCi ¼Xhi¼1hCjgijCi ¼

Xhi¼1

Xr

Xs

Xk

Xl

ZrCr;kCs;l

ZD


SðGÞ ¼1�

Phi¼1

Pr

Ps

Pk

Pl

ZrCr;kCs;l

RD


hPk

Zk

ð13aÞ

For infinite compact groups the symmetry measure is:

hCjgðaiÞjCiXr

Xs

ZrhFsjgijFri

¼Xr

Xs

Zr

ZD

F�s ð�xÞgðaiÞFrð�xÞd�x

¼Xr

Xs

Xk

Xl

ZrCr;kCs;l

ZD

f�l ð�xÞgðaiÞfkð�xÞd�x

hCjGjCi¼Yki¼1

ZBi

Ai

hCjgðaiÞjCidai

¼Xr

Xs

Xk

Xl

ZrCr;kCs;l

Yki¼1

ZBi

Ai

ZD

f�l ð�xÞgðaiÞfkð�xÞd�xdai

SðGÞ ¼1�

Pr

Ps

Pk

Pl

ZrCr;kCs;l

Qki¼1

RBi

Ai

RD

f�l ð�xÞgðaiÞfkð�xÞd�xdai

Pk

ZkQki¼1

RBi

Ai

dai

ð13bÞ

6.3 Wave-function example: the 2pz orbital of the hydrogen

atom

For the hydrogen atom analytical solutions exist, so we can

apply eqn (11a) and (11b). Here we consider the 2pz orbital of


hydrogen atom, quantum numbers: n = 2, l = 1, ml = 0

(we will work with atomic units): hr|Ci = Nexp(�12r)rcos y,

where N is the normalization factor, N ¼ 1ffiffiffiffiffiffi32pp .19 Rotation

around Z-axis by an angle of d is represented by the operator:

Rz(d): f- f+ d. It is clear that this symmetry operation only

changes the angle f. Since the function that represent the 2pzorbital depends only on r and y, rotation around the Z-axis by

any angle leaves the function unchanged: |Ci= Rz(a)|Ci, andtherefore we expect to get preservation of its full symmetry,

namely zero values for S(C2), S(Cn) and S(CN):

C2 : f! fþ p

Ck

n : f! fþ 2pnk

RzðdÞ : f! fþ d

hC2i ¼1

2hCjCi ½hCjEjCijhCjC2jC i�

¼ 1

2hCjCi ½hCjCi þ hCjCi� ¼ 1

hC2i ¼1

nhCjCiXnk¼1hCjCk

njCi

¼ 1

nhCjCiXnk¼1hCjCi ¼ nhCjCi

nhCjCi ¼ 1

hC1i ¼

R2p0

hCjRzðdÞjCidd

hCjCiR2p0

dd

¼

R2p0

hCjCidd

2phCjCi

¼hCjCi

R2p0

dd

2phCjCi ¼2phCjCi2phCjCi ¼ 1

And so, the symmetry measures are zero as expected:

S(C2) = 1 � hC2i = 0

S(Cn) = 1 � hCni = 0

S(CN) = 1 � hCNi = 0

That is, the 2pz orbital has CNv symmetry.

For the sake of demonstration, let us analyze now a

symmetry that the 2pz orbital is devoid of, namely inversion.

The inversion operator is: i(r,y,f) - (r,p � y,f + p) and we

operate it on the 2pz orbital function:

ı |Ci= N exp(�12r)rcos(p � y)

= N exp(�12r)r[cos y cos p � sin y sin p]

= N exp(�12r)r[(�1)cos y � 0sin y]

= �N exp(�12r)rcos y = �|Ci

ı |Ci= �|Ci

and the inversion measure is:

hCii ¼1

2hCjCi ½hCjEjCi þ hCjijCi�

¼ 1

2hCjCi ½hCjCi � hCjCi� ¼ 0

SðCiÞ ¼ 1� hCii ¼ 1

That is, we obtain the expected result that the 2pz orbital

function is totally devoid of inversion symmetry. In

contrast, however, the square of this function, which

represents the probability distribution of the electron state,

hr|C2i = N2exp(�r)r2cos2 y, still has rotational symmetry

around the Z-axis as before, but now has also inversion

symmetry. Let us show it:

ı|C2i = N2 exp(�r)r2cos2(p � y)

= N2 exp(�r)r2[cos y cos p � sin y sin p]2

= N2 exp(�r)r2[(�1)cos y � 0sin y]2

= �N2 exp(�r)r2cos2 y = �|C2i

ı|C2i = �|C2i

hCii ¼1

2hC2jC2i½hC2jEjC2i þ hC2jijC2i�

¼ 1

2hC2jC2i½hC2jC2i þ hC2jC2i� ¼ 1

SðCiÞ ¼ 1� hCii ¼ 0

And indeed, the square of the 2pz orbital has DNh

symmetry, a sub-group of which is inversion. Needless

to say, the same line of analysis follows for wave-functions

which are symmetry-distorted (say, by condensed-phase

interactions).

7. Conclusions: generalization and the chirality

measure

We introduced in this paper a novel general method for

calculating the continuous symmetry measures (CSM) both

for finite groups (eqn (5)) and for infinite compact groups

(eqn (6)). We showed how those equations are applicable

for calculating the symmetry content of vectors, matrices,

operators and functions. For each case we gave an introductory

physical example. The treatment of each of these mathematical

entities has been done for convenience of the various specific

applications. It is time now to generalize:

Any point in a given vector space can be written as a

linear combination of all the base elements of the space:

jCi ¼Pi

aijFii (where |Fii is the ith base element of the space

and ai is the coefficient of the ith base element in the linear

combination). When an operation is applied on the object the

result is another object in the same space, that can also be

written as a different linear combination of all the base

elements of the space: jOi ¼Pi

bijFii. Because the base

elements are the same, we can write this as: �B = gA. Here


A = (a1,a2,. . .,ai,. . .)T is a column vector containing the

coefficients of the original object, jCi ¼Pi

aijFii,�B = (b1,b2,. . .,bi,. . .)T is a column vector containing the

coefficients of the resulting object, jOi ¼Pi

bijFii and g is a

matrix representing the operation. In this way we can

transform any problem to a vector problem. On the other

hand, any vector in a finite dimensional space is of a function

(on a finite set), so we can also transform any problem to a

function problem. Also, instead of describing a molecule using

an assembly of different coordinate-vectors (as in section 3.2

and 3.4) we can describe it with one matrix, where each

column of the matrix is one of the vectors in the assembly.

In this case, we can apply the symmetry group operator for

matrices, and the result will be exactly the same as applying

the symmetry group operator for an assembly of different

coordinate-vectors.

This report focused on the CSM and not on the related

CCM (the continuous chirality measure).25 However, we

emphasize that in fact we also introduced here a general

solution to chirality measurements: We recall that chirality

implies a lack of improper symmetry, and therefore a chirality

measure can be based on estimating how close is an object to

having an improper symmetry, that is, how close is the object

from being achiral. The chirality measure in then the minimal

of all S(Sn) values (n = 1, 2, 4, 6. . .). One can use the general

equations in this paper to calculate the S(Sn) for vectors,

functions, operators, matrices and more. Because of the

central role of chirality in chemistry, biology and physics, we

shall treat it in detail in a separate report.

Appendix 1: the Bra-Ket notation

Following a general practice in chemistry and physics, we

employ the Bra-Ket notation, also known as Dirac notation.

However, for the sake of some other theoretical communities

where this notation is less common, here is a brief outline

of it:

Any point in a given vector space can be written as a linear

combination of all of the base elements of the space. The Ket,

|Ci, is a column vector containing the coefficients of the base

elements in the linear combination: |Ci = (C0,C1,C2,. . .)T.

Every Ket, |Ci, has a dual Bra, hC|, which is the conjugated

transpose (also termed the Hermitian conjugated) of the Ket,

and which is a row vector of the conjugated transpose of the

elements in the Ket: hC| = (C*0,C*1,C*2,. . .). The term hC|Oiis the inner product, the scalar multiplication, of objectsC and

O in the given space. The norm of the object |Ci (its size) isdefined by jCj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffihCjCi

p. If A: H - H is a linear

operator, we can apply A on the Ket, ||Ci to obtain a new

Ket ||Wi= (A|Ci). Composing the Bra, hC| with the operator

A results in the new Bra, (hC|A), defined as a linear functional

on H by the rule: (hC|A)|Oi = hC|(A|Oi). This expression is

commonly written as hC|A|Oi. If the same state vector appears

on both Bra and Ket side, this expression gives the expectation

value, or mean (average) value, of the observable represented

by operator A for the physical system in the state |Ci,written as hC|A|Ci. More detailed explanations can be found

in ref. 11.

Appendix 2: Deriving a that minimizes S(Ci)

of Fig. 2

0 ¼ @SðC iÞ@a

¼�2 cos a

2

� �sin a

2

� �1þ 2 sin2 a

2

� �� cos2 a

2

� �4 sin a

2

� �cos a

2

� �� 1þ 2 sin2 a

2

� �� 2� sinðaÞ � 1þ 2 sin2

a2

� �h i� 2 cos2

a2

� �sinðaÞ ¼ 0

� sinðaÞ 1þ 2 sin2a2

� �þ 2 cos2

a2

� �h i¼ 0

� 3 sinðaÞ ¼ 0

sinðaÞ ¼ 0! a ¼ 0; 180

a ¼ 0 ! SðC iÞ ¼ 1! max

a ¼ 180 ! SðC iÞ ¼ 0! min

Appendix 3: The D values of M1, M2

First, we move the center of mass of the systems to the origin.

To do so we calculate the shift parameter a:

a ¼

P4i¼1

P4j¼1

mi;j

P4i¼1

P4j¼1

1

¼ 26

16¼ 1:5:

Next, we subtract this value from each cell of the matrices:

W1 ¼M1 � a1 ¼

�0:5 �0:5 0:5 0:5�0:5 �0:5 0:5 0:5�0:5 �0:5 0:5 0:5�0:5 �0:5 0:5 0:5

0BB@

1CCA

W2 ¼M2 � a1 ¼

�0:5 0:5 �0:5 0:50:5 �0:5 0:5 �0:5�0:5 0:5 �0:5 0:50:5 �0:5 0:5 �0:5

0BB@

1CCA:

The S(Cs) and S(Ci) values for each matrix are:

M1 ! SðCsÞ ¼ 1� trðW�1 ðIW1I


�11 ÞÞ

2trðW�1 �W1Þ

¼ 1� 4þ 4

2 � 4 ¼ 0



�12 ÞÞ

2trðW�1 �W1Þ

¼ 1� 4� 4

2 � 4 ¼ 1

M2 ! SðCsÞ ¼ 1� trðW�2 ðIW2I


�11 ÞÞ

2trðW�2 �W2Þ

¼ 1� 4� 4

2 � 4 ¼ 1




�12 ÞÞ

2trðW�2W2Þ

¼ 1� 4þ 4

2 � 4 ¼ 0

Now we can calculate the D values:

D1 = S(Ci)S(Cs) = 1 � 0 = 0

D2 = S(Ci)S(Cs) = 1 � 0 = 0

These matrices have the minimal D value possible. This means

they are highly organized and have lower entropy compared to

the other possible arrangements.

Appendix 4: Deriving k that minimizes S(CN,z)

of Fig. 4

SðC1;zÞ ¼2ðp� 1ÞEHkþ 6pk2


0 ¼ @SðC1;zÞ@k

)

0 ¼ �12ð3p� 1ÞpEHk2 þ 12p2E2

Hkþ 2ðp� 1ÞE3H

ðpE2H � 4EHkþ 18pk2Þ2

6ð3p� 1Þk2 � 6pEHk� ðp� 1ÞE2H ¼ 0

k1;2 ¼6pEH � 2EH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27p2 � 24pþ 6p

36p� 12

k1 12:63! SðC1;zÞ ¼ 0:45! max

k2 �2:48! negative k is not physical

k ¼ 0! SðC1;zÞ ¼ 0! local min

Acknowledgements

We gratefully acknowledge Dr Mark Pinsky, Dr Pere

Alemany i Cahner, Dr. Hagit Hel-Or and Dr Jonathan Breuer

for reading the manuscript and contributing many useful

comments and insights. We thank Prof. Sason Shaik and Dr

David Danovich for their advice on issues of quantum

mechanics and wave functions. Supported by the Israel

Science Foundation (grant 307/08) and by the German-Israel

Binational Foundation (grant 983/2007).

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