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]
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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 journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 | 9655
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;
9656 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009
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.
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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:
9658 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009
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
�1ÞÞ þ trðW�3 ðU2W3U
�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
�1ÞÞ þ trðW�3 ðU1W3U
�11 ÞÞ
2trðW�3W3Þ
¼ 1� 4þ 0
2 � 4 ¼ 0:5
SðCiÞ ¼ 1� trðW�3 ðIW3I
�1ÞÞ þ trðW�3 ðU2W3U
�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
:
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 | 9659
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
trðH�0ðRzðdÞH0R�1z ðdÞÞÞ dd
¼ trðH�0 � H0ÞZ2p0
dd ¼ pE2H
2
Z2p0
trðH�1ðRzðdÞH0R�1z ðdÞÞÞ dd
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
pE2H � 4EHkþ 18pk2
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))).
9660 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009
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
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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
f�l ð�xÞgifkð�xÞ d�x
SðGÞ ¼1�
Phi¼1
Pr
Ps
Pk
Pl
ZrCr;kCs;l
RD
f�l ð�xÞgifkð�xÞ d�x
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
9662 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009
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
This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 | 9663
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
�1ÞÞ þ trðW�1 ðU1W1U
�11 ÞÞ
2trðW�1 �W1Þ
¼ 1� 4þ 4
2 � 4 ¼ 0
SðCiÞ ¼ 1� trðW�1 ðIW1I
�1ÞÞ þ trðW�1 ðU2W1U
�12 ÞÞ
2trðW�1 �W1Þ
¼ 1� 4� 4
2 � 4 ¼ 1
M2 ! SðCsÞ ¼ 1� trðW�2 ðIW2I
�1ÞÞ þ trðW�2 ðU1W2U
�11 ÞÞ
2trðW�2 �W2Þ
¼ 1� 4� 4
2 � 4 ¼ 1
9664 | Phys. Chem. Chem. Phys., 2009, 11, 9653–9666 This journal is �c the Owner Societies 2009
SðCiÞ ¼ 1� trðW�2 ðIW2I
�1ÞÞ þ trðW�2 ðU2W2U
�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
pE2H � 4EHkþ 18pk2
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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