and
, and can be labeled by two quantum numbers, l and m, related to
the corresponding eigenvalues. The complex spherical harmonic
can be written as [5,
6, 7]
| ECCC-3 Contents Page | THEOCHEM Home Page | Elsevier Chemistry Home Page |
Miguel A. Blanco1 and M. Flórez1and M. Bermejo2
1Departamento de Química Física y Analítica.
Facultad de Química.
Universidad de Oviedo. 33006-Oviedo. Spain.
2Departamento de Física. Facultad de Ciencias.
Universidad de Oviedo. 33007-Oviedo. Spain.
e-mail:
miguel@carbono.quimica.uniovi.es
HomePage:
http://www.uniovi.es/~quimica.fisica/qcg/qcghome.html
Rotation matrices (or Wigner D functions) are the matrix representations of the rotation operators in the basis of the spherical harmonics. They are the key entities in the generation of symmetry-adapted functions by means of projection operators. Although their expression in terms of ordinary (complex) spherical harmonics and Euler rotation angles is well known, an alternative representation using real spherical harmonics is desirable. The aim of this contribution is to obtain a general algorithm to compute the representation matrix of any point-group symmetry operation in the basis of the real spherical harmonics, paying attention to the use of recurrence relations that allow the treatment of functions with high angular momenta.
The choice of real spherical harmonics (RSH) as basis functions in electronic structure calculations has a number of advantages over other alternatives. Ordinary complex spherical harmonics are easier to manipulate theoretically, due to a number of useful relations that loose their simplicity when stated in terms of the RSH. However, they are complex functions requiring twice the computer memory needed by the RSH, as well as a complete rewriting of quantum-mechanical programs to diagonalize hermitian matrices instead of real ones.
A second alternative are the xiyjzk cartesian functions. These are real functions and are used in most Self Consistent Field (SCF) programs. Once again, the expressions of integrals in terms of these functions are simpler than those in terms of the RSH, due to their cartesian tensorial character. However, these functions have two main problems: their use becomes more involved for high angular momenta, and they include undesired atomic symmetry-adapted functions when the index l=i+j+k>2. Another problem associated to the use of these functions is their use along with Slater Type radial functions, since the best compilation of optimized exponents for Slater Type Orbitals (STO), that of Clementi and Roetti [1], uses spherical harmonics which can be considered to be either real or complex, due to the spherical symmetry of the free atoms.
In addition to the above mentioned advantages, the real spherical harmonics are the symmetry-adapted functions for atoms under most point-groups. In relation to this subject, McWeeny [2] points out: ``Although the complex functions provide the standard representations of the full group D3, the real spherical harmonics are often just the correct combinations to carry irreducible representations of its subgroups -- the point groups.'' Appendix 1 of the same reference also gives an exhaustive list of the symmetry-adapted functions of the crystal point-groups. The RSH are, therefore, most adequate basis functions for calculations in which atomic symmetry is important, when high angular momenta are needed, or when using high quality Slater Type radial basis functions.
In order to use the RSH the following three basic items are necessary: a definition suitable for recurrent evaluation of functions with high angular momenta, a set of rules for the evaluation of the coupling coefficients between them (Gaunt coefficients), and a mean of computing their rotation matrices in order to obtain symmetry functions. The first item is presented in Section 2, dedicated to the definition and basic properties of the RSH. The second item has been recently treated by Homeier and Steinborn [3]. The third item, obtaining in an efficient way the representation matrices of the symmetry operations in the basis of the RSH, is the main objective of this contribution, and will be addressed to in Section 3. Recently, Ivanic and Ruedenberg [4] have developed a completely different method with the same goal. These authors deal directly with the rotation matrices while, in the present work we deal with their representation in terms of the Euler angles.
Real spherical harmonics are defined in terms of their complex analogs.
These are the eigenfunctions of the orbital angular momentum operators,
and
, and can be labeled by two quantum numbers, l and m, related to
the corresponding eigenvalues. The complex spherical harmonic
can be written as [5,
6, 7]
where l takes non-negative integer values, the possible values for
m are the integers from -l to l,
and
are the angular spherical coordinates,
is the normalization factor
and
are the Legendre associated polynomials, which can be defined through the
Rodrigues formula,
From this definition one can derive [6]
where
means -m. Introducing the notation
for the angular coordinates, one can also derive the parity property
where
is the spatial inversion operator.
The complex dependence in
,
, is characteristic of the spherical symmetry of atoms, being the eigenfunction
of
. However, functions with different m are degenerate for effective
hamiltonians with spherical symmetry, so any linear combination of these
functions will still be an eigenfunction of this kind of hamiltonian. In
particular, one can choose real functions by combining complex conjugate
functions, corresponding to opposite values of m. In this way, the real
spherical harmonics are defined as [7]
The
factor has been introduced following Chisholm [7],
in order to obtain signless expressions for the real spherical harmonics (see
Table 1). These functions can also be
written as
by defining the azimuthal function as
The real spherical harmonics for
are presented in Table
1.
Table 1: Real spherical harmonics with
. The factors
have been omitted for simplicity.
To obtain the real spherical harmonics one can use closed formulas for
Legendre polynomials (see for example Ref. [5]),
and then evaluate directly
and
. However, this can lead to numerical instabilities, in addition to being
inefficient when, as is usually the case, all the RSH up to a given l
are needed. In this case, it is more efficient to use stable recurrence
relations, such as [8]:
For the
functions, one can start with the
and
values, using then the known trigonometric formulas:
With these recurrence relations, one can obtain the
and
functions separately, so that no one expression is computed more than once,
because only Legendre polynomials with
are needed, and
functions for a given m are common for all l values. The
normalization factors
(
) can be computed only once, at the very beginning of a calculation, and then
used any time a spherical harmonic is evaluated.
Given their definition, Eq. 6, the RSH have the following properties. First of all, since they always involve complex spherical functions with the same l value, they share the same symmetry under inversion,
Secondly, it is easy to prove their orthonormality from that of their complex counterparts,
Thirdly, they are real functions,
Eq. 6 can be written in matrix form as
where we define column vectors containing real and complex spherical harmonics, ordered by raising m values, and the transformation matrix between them for a given l is
This matrix can be represented by a set of simple rules:
In the last four rules, m>0 is assumed. It can also be proved
that
is a unitary matrix, that is,
In this way, the matrix definition can be reversed to obtain
Using the previous relation and the completeness of the complex functions, it is possible to demonstrate that the RSH constitute a complete basis set. Effectively, if a given function can be written as a linear combination of complex spherical harmonics of a given l, this can also be made in terms of the real ones, by making an appropriate transformation on the coefficients of the linear combination:
where
. An special case of the previous expression occurs when the
coefficients are in turn complex spherical functions of a different coordinate
system than that of the initial expansion. In this case, if
, then
which, being the RSH real functions, equals to
, and so the interesting property follows:
which is very useful in the Laplace expansion of the Coulomb repulsion
operator
.
The use of symmetry is fundamental for simplifying the quantum mechanical determination of the electronic structure. Be it spatial or point symmetry, its use can greatly reduce the number of non-equivalent integrals to compute, and also the size of matrices in the program, by employing symmetry-adapted functions. In order to obtain these symmetry functions, it is necessary to know the transformation properties of the atomic basis functions under symmetry operations. In this work, we will restrict ourselves to point symmetry groups, because they are the only ones needed for finite molecules.
First of all, let us see how point symmetry operations are represented.
Let
be a symmetry operation of the G point group, and
a point in the three-dimensional space. If we apply the symmetry operation to
this point, we obtain a new point
which relates to the original one through
where
is the matrix associated to the operation, that is, the representation matrix
of the
operation in the basis of the cartesian coordinates. The operations are
classified as proper or improper depending upon the determinant of their
associated matrices: +1 and -1, respectively. Any improper operation can be seen
as the product of a proper operation and the inversion:
where
is an improper operation and
is the corresponding proper operation. Remembering that the representation of a
product of operations is the product of the representations of each operation,
, and the
matrix is just the negative of the unit matrix, one can see that
.
Up to this point, all representations are taken in the cartesian basis. If we want to obtain the representation matrices in the basis of the real spherical harmonics, we must know what happens to these functions upon the symmetry transformation. It can be shown that, for point symmetry operations,
that is, the spherical harmonics of a given l are transformed into a
linear combination of functions of the same l. It must be noted that, in
this definition, the matrix
multiplies the spherical harmonics as row vectors, whereas the rotation
matrices for cartesian coordinates were defined multiplying column vectors (Eq.
25). This is the usual convention in
both cases, and it should be remembered when relating them, as well as when
obtaining the representation matrices of projection operators.
is the representation matrix of the symmetry operation
in the basis of the RSH of order l, and it can be symbolically
expressed as
An important special case of representation matrix is that of the inversion. Remembering Eq. 15, we can write it as
that is,
times the unit matrix. In this way, the matrix representation of an improper
operation can be obtained as a function of the matrix of its corresponding
proper operation as:
Taking into account that proper symmetry operations are equivalent to rotations in the three-dimensional space, the problem of obtaining the representations of point-group symmetry operations is reduced to the obtaining of the representation of any rotation operation. The representation matrices of rotation operations are generically called rotation matrices of the selected basis set.
Usually, point-group symmetry operations are specified by their matrix
representations in the basis of the cartesian coordinates, so we will focus in
the obtaining of the representation matrices of the RSH taking those of the
cartesian coordinates as a starting point. This will be specially useful in
crystalline applications, where parallel coordinate systems are best suited for
integral calculation, symmetry operations do not necessarily coincide with
cartesian axis, and the coordinate rotation matrices can be easily obtained from
those of the space group. In order to relate both representations, it must be
noted that
functions transform in the same way as the cartesian coordinates under
point-group operations, and so
where
is the matrix representation of the
operator in the cartesian basis. This relation gives us the matrix
representation of the operation in the basis of p functions. Since the
s spherical harmonic is a constant, it is invariant under any symmetry
operation, and so
for any
. These two matrices, for s and p functions, will serve as the
starting point of the recurrence relations that will allow us to obtain the
rotation matrices for the RSH of any order.
The traditional method for obtaining the representation matrices is to represent the functions, then apply the operation, and finally try to get the elements of the matrix by inspection. It is clear that this is not a computationally viable method, and so we must try to develop better ones. A similar method would be to obtain the RSH as a polynomial in cartesian coordinates, and substitute the rotation matrix for them. However, this is still an inefficient method. Another possible method would be to obtain the rotation matrices of complex spherical harmonics, and then use the transformation matrices between real and complex functions:
where
is the rotation matrix of complex spherical harmonics, also known as Wigner
D function [5]. Writing the above
expression in matrix form,
we make clear the transformation between rotation matrices. This is a better
method than the preceding ones, but it still has a big drawback: it involves
complex matrices in the intermediate steps of the calculation of
matrices, which are real by definition.
Figure 1: Description of an arbitrary rotation in terms of
Euler angles.
The Wigner D functions are well known in the quantum theory of angular momentum [5], and their properties can be easily found in the literature. One of their most important properties is their expression in terms of Euler angles (see Fig. 1 for their definition): since any given rotation can be expressed in terms of these angles, the rotation matrix can be made dependent on them through
The fact that
matrices are real makes viable a new scheme for obtaining the rotation matrices
. The
matrix can be put in closed form as [5]:
where k runs through all integer values for which the factorials
involved exist. However, as it happened with the spherical harmonics, it is more
interesting to obtain this matrix by means of recurrence relations. To
accomplish this task, the following relations may be of use (the
dependence is omitted for the sake of simplicity):
In addition, the following expressions for special values of the indices will be needed:
The elements of
for all values of the indices can then be computed by using the above
expressions recursively, as we shall see below.
The scheme for obtaining the matrices
is as follows. First, one substitutes the definition of the
matrix (Eq. 36) in Eq.
35, studying the different cases that
appear due to the indices of the
matrices. With these, expressions for
in terms of
and the Euler angles will be obtained. Then we will see how to obtain the
trigonometric functions of Euler angles in terms of the
matrix elements. In this way, we can compute
and
matrices, which combined with a set of recurrence rules allow us to obtain the
rest of the
matrices, and with them the
matrices, which constitute the aim of this work.
Eq. 34 will be the starting
point of the following discussion. First, we must note that, due to the
properties of
, whose only non-zero elements are those in the two diagonals, each summation
through
and
has two elements at most, giving a total maximum of four terms. So, it is
affordable to particularize these elements and try to find simpler closed
expressions that do not involve complex numbers. There are four possible cases,
depending on which indices are zero (
):
After replacing in the previous expressions the definition of
matrices (Eq. 36), the sorting of
different cases for the
matrices, and a certain amount of algebra, the following relation is obtained (
):
where definition in Eq. 8 has been
used, and
. In this way, knowing the
matrices and the Euler angles, the
matrices can be obtained. In fact, it suffices with half the
matrices, since only elements with
are needed. It is possible to reduce even more this requirements by using the
mirror symmetry of both diagonals of these matrices, so that the lower of the
four triangles that the diagonals define is only needed.
Now we will obtain the starting elements for the recurrence relations.
These will be the elements of the lower triangle of the
and
matrices, that is, the
,
,
,
and
elements. The first one is the simplest,
which makes
, since the s function does not change upon rotations. To obtain the
element, we use Eq. 42:
Eq. 41 is used for the remaining three elements:
The above expressions can be introduced in Eq.
47 to obtain the
matrix:
Relating this matrix to its expression in terms of the rotation matrix in
the cartesian basis, Eq. 31, it is possible
to obtain the trigonometric functions of the Euler angles. Certainly, given
, it can be established that
so that
. In the same way, the original elements of the
matrix,
,
, and
, can be obtained. When
(
), the following relations can be found:
However, when
these relations fail. In this case, the
matrix can be written as
so that there is a single independent rotation angle,
. Arbitrarily choosing
, we find:
when
(
). This particular case corresponds to rotations about the z axis. Since
it can introduce problems in the application of recurrence relations, we will
consider it separately in some cases. In this way, the trigonometric functions
of Euler angles are obtained from
matrices for a given rotation, along with the starting elements of the
matrix.
To continue, we need the particular recurrence relations to use in the
generation of
matrices. The first and more important rule is that relating the l
order matrix with those of orders l-1 and l-2, Eq.
39, which can be written in a more useful form
as
This relation gives access to all elements of the matrix of a given order once the previous two orders are known, with the exception of the first and last two rows and columns. The problem can be reduced to the computation of the last two rows by using the symmetry relations in Eq. 38. To evaluate these elements by means of recurrence rules, Eq. 41 can be used to obtain
In the same way, Eq. 42 can be transformed into
To obtain the rest of the elements of the last two rows, we could use the general recurrence relation Eq. 40. However, this requires two previous elements for each new one, making unnecessarily complicated the obtaining of these two rows: since there are particular expressions for them, Eqs. 41 and 42, it is more efficient to use such expressions to obtain descending recurrence rules for both. Thus, Eq. 65 can be used as the starting point for the last row, and then successively lower the second index by means of
which can be obtained from Eq. 41, and where
. The same can be done starting from Eq. 66,
stepping down through the second to last row by means of
obtained from Eq. 42. Since
, these recurrence relations are not valid. However, in this case Eqs.
41 and 42 can be
used to prove that all elements of the last two columns are zero except
and
.
Collecting all the previous relations, the following algorithm to obtain
matrices can be devised:
obtain
.
obtain trigonometric functions of
from
.
obtain
,
,
and
from
.
for
obtain
(
;
)
using Eq. 64.
obtain
for
(Eq. 67).
obtain
for
(Eq. 68).
end-for
In this way all the elements of the lower triangle of the
matrices of all orders can be computed. To obtain the rest of the elements with
m>0, the mirror properties through the diagonals (Eq.
38) can be used, and so all necessary elements
in Eq. 47 will be available. Since we
also have the sine and cosine of the angles
and
, along with the recurrence relations, Eqs. 13
and 14, we are able to evaluate the rotation
matrices of the RSH,
, for any order. Remembering that the representation matrix of the inversion
operation is the unit matrix times
, the representation matrices of any point symmetry operation in the basis of
the real spherical harmonics can be obtained from its representation matrix in
the cartesian basis.
One of us (MAB) is indebted to the Spanish Ministerio de Educación y Ciencia for a postgraduate grant. Financial support was provided by the Spanish DGICyT of the Ministerio de Educación y Ciencia under grant PB93-0327.
(Eq. 
(Eq.