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Exploiting time-reversal symmetry in ligand-field calculations
Volume182,number2
CHEMICALPHYSICSLETTERS
26 July 1991
Exploiting time-reversal symmetry in ligand-field calculations Mark J. Riley Institutjir
anorganische
undphysikaluche
Chemie,
Universitiit
Bern, Freiestrasse
3, CH-3000
Bern 9, Swrtzerland
Received16April I99I
It is shown that for all point groups which contain Cz or C, as an invariant subgroup, the ligand-field eigenvalue problem for odd-electron systems can be reducedto half the size. The procedure is a consequence of the way the eigenfunctions transform under time reversal.
1. Introduction
CPU time by factors of approximately respectively.
Group theory is a powerful tool for simplifying ligand-field calculations using the geometric symmetry of the molecule. Invariance of the molecular Hamiltonian to symmetry operations allows tensor operator techniques to construct a block-diagonal matrix for the eigenvalue problem. The time-reversal symmetry in Kramers’ degeneracy, however, has only recently been exploited [ 11. Kramer? degeneracy occurs in molecules with an odd number of electrons and causes all energy levels to have an evenfold degeneracy in the absence of a magnetic field [ 21. However, as there exists a relationship between the Kramers doublet wavefunction together with the (at least) twofold degenerate energy levels, there is clearly redundant information contained within the full problem. Riisch [ I ] presented a scheme whereby an eigenvalue problem of a general Hamiltonian which has only time-reversal symmetry could be reduced from diagonalizing a 2Nx2N complex matrix to the diagonalization of an NX N quaterionic matrix. This represents a computational advantage as both the storage space and CPU time are typically reduced by a factor of two. In the present work, it is pointed out that in the IJ, MJ) basis commonly used in the ligand-field problem of d” and f” systems, there exists a simple way to reduce the general 2N X 2N complex ligand-field matrix to a single NX N complex matrix for all point groups which contain C2 or C, as an invariant subgroup. This then reduces the storage and 0009-2614/91/$
03.50
0 1991 -
4 and 8,
2. Time-reversal symmetry and the quaterionic
structure For the odd-electron systems using d or f orbital bases, the basis size is necessarily even. If a basis contains the function I$, it also contains the time-reversed partner KqJ where K is the time-reversal operator [ 31. An eigenfunction of the molecular Hamiltonian, expressed as a linear combination of the basis functions, v= kg,
ak4k
+ bk&k
,
(la)
will have the other component of the Kramers doublet given by
k=l
where ak and bk are complex numbers. It has been shown [ 1] that a matrix evaluated in a basis consisting of N Kramers’ pairs, can be transformed into a quaterionic matrix where a matrix element is given by the quaterion, / &,=rtsittj+uk, (2) where i, j, k obey iZ=j2=k2=ijk= -1, and r, s, t, u are real numbers related to the original matrix elements by
Elsevier Science Publishers B.V. (North-Holland )
187
Volume 182, number 2
CHEMICAL PHYSICS LETTERS
rtsi=(nJHlm)=(KnlHlKm)*,
(3a)
r-ui=(KrtlHlm)=-(nlHIKm)*,
(3b)
The NxNmatrix Q can be diagonalized by adapting standard algorithms to quaterionic algebra, and this directly yields the eigenvalues. The Kramers doublets v, I+?are related to an eigenvector wq of Q by eqs. (1) and
26 July 1991
+Ky
=p1
‘+J-MJc:Icy, I, L, S,J, -MJ)
{K$}: bk=tL-uki.
(4)
It can be seen that if matrix elements between the two sets of basis functions {$l and {@I are zero in eq. (3b), then the quaterionic matrix in (2 ) reduces to a complex Hermitian matrix. The resulting Kramers doublet v/, p given by (I) has the bk coefficients zero, so that each component of the doublet is a linear combination of only one of the two sets of basis functions {$) or {@}:
k=l
a$@,.
(5)
These two sets, {@}and {Kg}, then make the ~Nx 2N matrix block diagonal into two separate NxN matrices. This blocking, however, does not correspond to a geometric symmetry and only one of the two matrices needs to be diagonalized, giving the energy levels and one component of all Kramers doublets. The other component can be generated, if necessary, using eq. ( lb). It remains to show under what conditions the matrix elements between the sets {$} and {K@}are zero.
3. Ligand-field theory The many-electron wavefunction ICV,1, L, S, J, M,), following the nomenclature used in ref. [4], provides a convenient basis for ligand-field calculations. An eigenfunction for odd-electron systems in terms of this basis is related to its Kramers’ partner by [31 V= T
188
c,I~,LL~,J,~J)
,
(6)
where I is 2 for d and 3 for f electron systems. For each weak-field multiple& J is a half-odd integer and the 2J+ 1 values of MJ= -J...- 512, - 312, - l/2, l/2,3/2, ...J. can be divided into one of the two sets: {p}: {M,=...-j/2,
ak=rktSki,
,
{M,=...-312,
-l/2,
...3/2. ...}.
l/2 ,... 512 ,... }.
(7)
These two sets are not connected by electron repulsion or spin-orbit coupling matrix elements as these are diagonal in the MJ quantum number. The ligand field can be expressed in terms of an expansion in spherical harmonics, a general matrix element of which is given by [4]
(8) The first 35 symbol, with the parity of the integral, restricts the values of k to 0, 2, 4 for d and 0, 2, 4, 6 for f electrons and q=O, . ... &k. The second 3i symbol in (8) gives the condition for the ligand-field matrix to be non-zero. Therefore, since the two sets of basis functions in (7 ) above have A4, -My = f 1, f. 3, ...) then q must be odd for functions of each set to be connected by the ligand field. If Ykqis zero for odd values of 4, the eigenfunctions of the Hamiltonian will take the form given in (5 ) and the matrix can be split. Components of the ligand field will be zero if they do not transform as totally symmetric in the point group of the molecular Hamiltonian. From examining correlation tables [ 31, it can be seen that the Ykqcomponent of the ligand field will be totally symmetric for odd q in the crystallographic point groups q=d!fJ-‘idy
Cl, Ci, G and G,. Alternatively, the coefficients of the spherical harmonic expansion of the ligand field have been given in terms of the complex one-electron d [ 41 and f [ 5 ] orbitals. In both cases, the coefficients with odd q contain only the one-electron matrix elements ( ml1 V)rnr > when m,- ml is odd. For d orbitals, this will occur only when members of the two sets,
Volume182,number 2
d,: {z2, x2-y’,
CHEMICALPHYSICSLETTERS
xyj , d,: {xz, yz) ,
(9)
transform as different irreducible representations, since these orbitals are constructed from combinations of the complex orbitals with even and odd values of m, in each set, respectively. In the case off electrons, the two sets are f,: {23,2(x2-yZ),xyz} , f,: {x(yZ-z2),y(z2-X2),
x3,$},
(10)
which are again constructed from combinations of the complex orbitals with even and odd values of m,, respectively. Inspection of character tables will show how these basis functions transform [ 61. Members of the two sets in both d and f electron cases transform as different irreducible representations for all crystallographic point groups except C,, Ci, C,, and C3i-
This is easy to see from considering the CZ and C, point groups. In the C2 point group d, -A-d, -B where - denotes “transforms as the irreducible representation”. Similarly f, -A N f, - B in the Cz point group. In the C, point group d, -A’ -d, -A” and f, - A” nvf, - A’. In all point groups which have a CZ and/or C, as a subgroup, the molecular axes can be defined such that the matrix element in (8) is zero for odd q. From subgroup diagrams [3,7], this includes all crystallographic point groups other than C,. Ci( =S,), C3, and C3i( ES,). Consideringall molecular point groups, one concludes that this corresponds to all groups excepting C, and S2,, for n odd. For all these point groups, the odd-electron ligandfield problem of diagonalizing a 2Nx 2N matrix can be reduced to that of diagonalizing an NX N matrix, by virtue of the fact that the energies and wavefunctions need to be calculated for only one of the Kramers doublet components. It should be noted that for some point groups, this requires an unconventional axes system. For example, in D3 the z axis should be defined as one of the c2 axes so that C, is an invariant subgroup of D,. Similarly for the Cj, point group, the z axis must be perpendicular to one of the uy planes so that C, is an invariant subgroup of C3”.
26 July 1991
4. Group theory The irreducible representation (hereafter: irrep), r, of a particular point group consists of a set of matrices, T(R), of order II- for the R symmetry operations of the group. The trace of these matrices gives the characters T(R) of irrep I-. It has been shown that the state(s) Kv/ which are time-reversed degenerate with the state(s) v/ of irrep I- will transform as r* [ 3,7]. There are then three cases that can be considered to classify the behaviour of a state I and its partner Kv/ for odd-electron systems [ 3,7 1: (a) T(R)=T(R)*, T(R)=T(R)*, the matrices and characters are real. If v/ is a basis for r, then KI,U must also be a basis for r. (b) T(R)#F(R)*, T(R)#I’(R)*, both matrices and characters are complex. If y/is a basis for r, then KY is a basis for I-’=r*. (c) T(R)#T(R)*, F(R)=T(R)*, the matrices are complex (symplectic) [ 31, but the characters are real. The order of the irrep is >2..The group has operations which transform w and KY as parts of the same irrep I-. This occurs for real irreps of non-Abelian point groups. This classification is that same as given by Koster et al. [ 71 for odd-electron systems. In case (a), the two states v/ and KY would be expected on spatial symmetry alone to be nondegenerate as they transform as the same irrep, but their degeneracy is due to time-reversal symmetry. Basis functions belonging to each of the two sets (0) and {Q} in (7) will also transform as the same irrep in this case. They will interact and the general mixed wavefunctions given in (1) will result, rather than the form desired in eqs. (5). In case (b), however, the sets (0) and {K@}will transform as different irreps and the resulting eigenfunctions will be of the form given by eqs. (5). Although in case (c ), y and Ku/ transform as the same irrep, by a suitable choice of axes they can still be made diagonal within this representation. This can be found by lowering the symmetry of the group G which contains a F of type (c) until it is of type (a) or (b). Using correlation tables [ 71, one finds that if the axes system of G is defined such that the principal axis has a c2 or a,, symmetry element, all irreps of type (c) then reduce to those of type (b). The only point groups where the matrix-halving approach will 189
Volume 182, number 2
CHEMICAL PHYSICS LETTERS
not work are those which contain irreps of type (a). As found before, these point groups are C, and S2n (n odd).
26 July 1991
Acknowledgement I would like to thank H.U. Giidel for support and encouragement. This work was supported by the Swiss National Science Foundation.
5. Conclusions It has been shown that in odd-electron ligand-field calculations for certain point groups, only half of the weak field IJ, M/> basis needs to be used. These point groups are those which have C2 and/or C, as an invariant subgroup which will be all point groups except C, and S,, for odd n. This applies to both d and felectrons and, assuming the time to diagonalize a matrix is proportional to the dimension cubed [ 11, this represents a computational saving of a factor of 8 over diagonalking the matrix with a full basis.
190
References [I] N. R&h, Chem. Phys. 80 (1983) 1. [ 21 C.J. Ballhausen, Introduction to ligand field theory (McGrawHill, New York, 1962). 131 P.H. Butler, Point group symmetry applications (Plenum Press, New York, 1981 ). [4] M. Gerloch, Magnetism and ligandfield analysis (Cambridge Univ. Press, Cambridge, 1983). [5] W. Urland, Chem. Phys. 14 (1976) 393. [ 6 1D.C. Harris and M.D. Bertolucci, Symmetry and spectroscopy (Oxford Univ. Press, Oxford, 1978). [7]G.F. Koster, J.O. DImmock, R.G. Wheeler and H. Statz, Properties of the thirty-two point groups [MIT Press, Cambridge, MA, 1963).
CHEMICALPHYSICSLETTERS
26 July 1991
Exploiting time-reversal symmetry in ligand-field calculations Mark J. Riley Institutjir
anorganische
undphysikaluche
Chemie,
Universitiit
Bern, Freiestrasse
3, CH-3000
Bern 9, Swrtzerland
Received16April I99I
It is shown that for all point groups which contain Cz or C, as an invariant subgroup, the ligand-field eigenvalue problem for odd-electron systems can be reducedto half the size. The procedure is a consequence of the way the eigenfunctions transform under time reversal.
1. Introduction
CPU time by factors of approximately respectively.
Group theory is a powerful tool for simplifying ligand-field calculations using the geometric symmetry of the molecule. Invariance of the molecular Hamiltonian to symmetry operations allows tensor operator techniques to construct a block-diagonal matrix for the eigenvalue problem. The time-reversal symmetry in Kramers’ degeneracy, however, has only recently been exploited [ 11. Kramer? degeneracy occurs in molecules with an odd number of electrons and causes all energy levels to have an evenfold degeneracy in the absence of a magnetic field [ 21. However, as there exists a relationship between the Kramers doublet wavefunction together with the (at least) twofold degenerate energy levels, there is clearly redundant information contained within the full problem. Riisch [ I ] presented a scheme whereby an eigenvalue problem of a general Hamiltonian which has only time-reversal symmetry could be reduced from diagonalizing a 2Nx2N complex matrix to the diagonalization of an NX N quaterionic matrix. This represents a computational advantage as both the storage space and CPU time are typically reduced by a factor of two. In the present work, it is pointed out that in the IJ, MJ) basis commonly used in the ligand-field problem of d” and f” systems, there exists a simple way to reduce the general 2N X 2N complex ligand-field matrix to a single NX N complex matrix for all point groups which contain C2 or C, as an invariant subgroup. This then reduces the storage and 0009-2614/91/$
03.50
0 1991 -
4 and 8,
2. Time-reversal symmetry and the quaterionic
structure For the odd-electron systems using d or f orbital bases, the basis size is necessarily even. If a basis contains the function I$, it also contains the time-reversed partner KqJ where K is the time-reversal operator [ 31. An eigenfunction of the molecular Hamiltonian, expressed as a linear combination of the basis functions, v= kg,
ak4k
+ bk&k
,
(la)
will have the other component of the Kramers doublet given by
k=l
where ak and bk are complex numbers. It has been shown [ 1] that a matrix evaluated in a basis consisting of N Kramers’ pairs, can be transformed into a quaterionic matrix where a matrix element is given by the quaterion, / &,=rtsittj+uk, (2) where i, j, k obey iZ=j2=k2=ijk= -1, and r, s, t, u are real numbers related to the original matrix elements by
Elsevier Science Publishers B.V. (North-Holland )
187
Volume 182, number 2
CHEMICAL PHYSICS LETTERS
rtsi=(nJHlm)=(KnlHlKm)*,
(3a)
r-ui=(KrtlHlm)=-(nlHIKm)*,
(3b)
The NxNmatrix Q can be diagonalized by adapting standard algorithms to quaterionic algebra, and this directly yields the eigenvalues. The Kramers doublets v, I+?are related to an eigenvector wq of Q by eqs. (1) and
26 July 1991
+Ky
=p1
‘+J-MJc:Icy, I, L, S,J, -MJ)
{K$}: bk=tL-uki.
(4)
It can be seen that if matrix elements between the two sets of basis functions {$l and {@I are zero in eq. (3b), then the quaterionic matrix in (2 ) reduces to a complex Hermitian matrix. The resulting Kramers doublet v/, p given by (I) has the bk coefficients zero, so that each component of the doublet is a linear combination of only one of the two sets of basis functions {$) or {@}:
k=l
a$@,.
(5)
These two sets, {@}and {Kg}, then make the ~Nx 2N matrix block diagonal into two separate NxN matrices. This blocking, however, does not correspond to a geometric symmetry and only one of the two matrices needs to be diagonalized, giving the energy levels and one component of all Kramers doublets. The other component can be generated, if necessary, using eq. ( lb). It remains to show under what conditions the matrix elements between the sets {$} and {K@}are zero.
3. Ligand-field theory The many-electron wavefunction ICV,1, L, S, J, M,), following the nomenclature used in ref. [4], provides a convenient basis for ligand-field calculations. An eigenfunction for odd-electron systems in terms of this basis is related to its Kramers’ partner by [31 V= T
188
c,I~,LL~,J,~J)
,
(6)
where I is 2 for d and 3 for f electron systems. For each weak-field multiple& J is a half-odd integer and the 2J+ 1 values of MJ= -J...- 512, - 312, - l/2, l/2,3/2, ...J. can be divided into one of the two sets: {p}: {M,=...-j/2,
ak=rktSki,
,
{M,=...-312,
-l/2,
...3/2. ...}.
l/2 ,... 512 ,... }.
(7)
These two sets are not connected by electron repulsion or spin-orbit coupling matrix elements as these are diagonal in the MJ quantum number. The ligand field can be expressed in terms of an expansion in spherical harmonics, a general matrix element of which is given by [4]
(8) The first 35 symbol, with the parity of the integral, restricts the values of k to 0, 2, 4 for d and 0, 2, 4, 6 for f electrons and q=O, . ... &k. The second 3i symbol in (8) gives the condition for the ligand-field matrix to be non-zero. Therefore, since the two sets of basis functions in (7 ) above have A4, -My = f 1, f. 3, ...) then q must be odd for functions of each set to be connected by the ligand field. If Ykqis zero for odd values of 4, the eigenfunctions of the Hamiltonian will take the form given in (5 ) and the matrix can be split. Components of the ligand field will be zero if they do not transform as totally symmetric in the point group of the molecular Hamiltonian. From examining correlation tables [ 31, it can be seen that the Ykqcomponent of the ligand field will be totally symmetric for odd q in the crystallographic point groups q=d!fJ-‘idy
Cl, Ci, G and G,. Alternatively, the coefficients of the spherical harmonic expansion of the ligand field have been given in terms of the complex one-electron d [ 41 and f [ 5 ] orbitals. In both cases, the coefficients with odd q contain only the one-electron matrix elements ( ml1 V)rnr > when m,- ml is odd. For d orbitals, this will occur only when members of the two sets,
Volume182,number 2
d,: {z2, x2-y’,
CHEMICALPHYSICSLETTERS
xyj , d,: {xz, yz) ,
(9)
transform as different irreducible representations, since these orbitals are constructed from combinations of the complex orbitals with even and odd values of m, in each set, respectively. In the case off electrons, the two sets are f,: {23,2(x2-yZ),xyz} , f,: {x(yZ-z2),y(z2-X2),
x3,$},
(10)
which are again constructed from combinations of the complex orbitals with even and odd values of m,, respectively. Inspection of character tables will show how these basis functions transform [ 61. Members of the two sets in both d and f electron cases transform as different irreducible representations for all crystallographic point groups except C,, Ci, C,, and C3i-
This is easy to see from considering the CZ and C, point groups. In the C2 point group d, -A-d, -B where - denotes “transforms as the irreducible representation”. Similarly f, -A N f, - B in the Cz point group. In the C, point group d, -A’ -d, -A” and f, - A” nvf, - A’. In all point groups which have a CZ and/or C, as a subgroup, the molecular axes can be defined such that the matrix element in (8) is zero for odd q. From subgroup diagrams [3,7], this includes all crystallographic point groups other than C,. Ci( =S,), C3, and C3i( ES,). Consideringall molecular point groups, one concludes that this corresponds to all groups excepting C, and S2,, for n odd. For all these point groups, the odd-electron ligandfield problem of diagonalizing a 2Nx 2N matrix can be reduced to that of diagonalizing an NX N matrix, by virtue of the fact that the energies and wavefunctions need to be calculated for only one of the Kramers doublet components. It should be noted that for some point groups, this requires an unconventional axes system. For example, in D3 the z axis should be defined as one of the c2 axes so that C, is an invariant subgroup of D,. Similarly for the Cj, point group, the z axis must be perpendicular to one of the uy planes so that C, is an invariant subgroup of C3”.
26 July 1991
4. Group theory The irreducible representation (hereafter: irrep), r, of a particular point group consists of a set of matrices, T(R), of order II- for the R symmetry operations of the group. The trace of these matrices gives the characters T(R) of irrep I-. It has been shown that the state(s) Kv/ which are time-reversed degenerate with the state(s) v/ of irrep I- will transform as r* [ 3,7]. There are then three cases that can be considered to classify the behaviour of a state I and its partner Kv/ for odd-electron systems [ 3,7 1: (a) T(R)=T(R)*, T(R)=T(R)*, the matrices and characters are real. If v/ is a basis for r, then KI,U must also be a basis for r. (b) T(R)#F(R)*, T(R)#I’(R)*, both matrices and characters are complex. If y/is a basis for r, then KY is a basis for I-’=r*. (c) T(R)#T(R)*, F(R)=T(R)*, the matrices are complex (symplectic) [ 31, but the characters are real. The order of the irrep is >2..The group has operations which transform w and KY as parts of the same irrep I-. This occurs for real irreps of non-Abelian point groups. This classification is that same as given by Koster et al. [ 71 for odd-electron systems. In case (a), the two states v/ and KY would be expected on spatial symmetry alone to be nondegenerate as they transform as the same irrep, but their degeneracy is due to time-reversal symmetry. Basis functions belonging to each of the two sets (0) and {Q} in (7) will also transform as the same irrep in this case. They will interact and the general mixed wavefunctions given in (1) will result, rather than the form desired in eqs. (5). In case (b), however, the sets (0) and {K@}will transform as different irreps and the resulting eigenfunctions will be of the form given by eqs. (5). Although in case (c ), y and Ku/ transform as the same irrep, by a suitable choice of axes they can still be made diagonal within this representation. This can be found by lowering the symmetry of the group G which contains a F of type (c) until it is of type (a) or (b). Using correlation tables [ 71, one finds that if the axes system of G is defined such that the principal axis has a c2 or a,, symmetry element, all irreps of type (c) then reduce to those of type (b). The only point groups where the matrix-halving approach will 189
Volume 182, number 2
CHEMICAL PHYSICS LETTERS
not work are those which contain irreps of type (a). As found before, these point groups are C, and S2n (n odd).
26 July 1991
Acknowledgement I would like to thank H.U. Giidel for support and encouragement. This work was supported by the Swiss National Science Foundation.
5. Conclusions It has been shown that in odd-electron ligand-field calculations for certain point groups, only half of the weak field IJ, M/> basis needs to be used. These point groups are those which have C2 and/or C, as an invariant subgroup which will be all point groups except C, and S,, for odd n. This applies to both d and felectrons and, assuming the time to diagonalize a matrix is proportional to the dimension cubed [ 11, this represents a computational saving of a factor of 8 over diagonalking the matrix with a full basis.
190
References [I] N. R&h, Chem. Phys. 80 (1983) 1. [ 21 C.J. Ballhausen, Introduction to ligand field theory (McGrawHill, New York, 1962). 131 P.H. Butler, Point group symmetry applications (Plenum Press, New York, 1981 ). [4] M. Gerloch, Magnetism and ligandfield analysis (Cambridge Univ. Press, Cambridge, 1983). [5] W. Urland, Chem. Phys. 14 (1976) 393. [ 6 1D.C. Harris and M.D. Bertolucci, Symmetry and spectroscopy (Oxford Univ. Press, Oxford, 1978). [7]G.F. Koster, J.O. DImmock, R.G. Wheeler and H. Statz, Properties of the thirty-two point groups [MIT Press, Cambridge, MA, 1963).