Derivation of Hessian in symmetry co ordinates
(App endix to: Van Vlijmen and Karplus: Normal mo de analysis of large systems with
icosahedral symmetry: Application to (Dialanine) in full and reduced basis set implemen-
60
tations)
Let V be the mass-weighted Hessian matrix of the p otential energy of the symmetric
multimer. Within the harmonic approximation the total potential energy U of the system
is a quadratic form in Cartesian displacement co ordinates fr ; i =1;n N g, relative to the
i G
energy minimum; n is the numb er of monomers and N is three times the numb er of atoms
G
of the monomer:
X
1
U (r)= v r r ; (A1)
ij i j
2
i;j
where v is the i; j th entry of V. The orthogonal co ordinate transformation to symmetry
ij
co ordinates preserves the quadratic form:
X
1
l;s ls;
;
(r)q (r); (A2) q U (r)=
k k;
2
k l s;
where k and are summed to N , the summations over l and are over all irreps (including
rep eats of the same irrep typ e), and the summations over s and are over the dimension s
p
ls;
of each irrep; is the entry of the transformed Hessian that corresp onds to co ordinates
k;
l;s
;
q and q , and denotes the complex conjugate.
k
The total numb er of terms in Eqns. A1 and A2 is the same. Further, the total p otential
0
energy of the system do es not change under rotations g 2 G, since the distances between
atoms do not change. This prop erty imp oses limits on the p ossible values of co ecients
ls;
. The pro of of the blo ck-diagonal form of the Hessian in symmetry co ordinates that
k;
2
is given here is based on the discussion by Lyubarskii.
0
After transformation of the Cartesian co ordinates by the symmetry op eration g (result-
0 0
0 0
= T (g )r; T (g ) is the transformation matrix), we , where r ing in co ordinate vector r
g g
write the expression for U as:
X
1
l s; l;s
;
0 0 0
q (r )q U (r )= (r ) (A3)
g g g
k; k
2
k l s; 1
The symmetry co ordinates after rotation are given by:
l;s l;s l;s
0
0 0
q (r )=r = T (g )r
g g
k k k
s
l
X
l;s l;m
0 l 0
= r T (g ) r (g )
sm
k k
m=1
s
l
X
l;m
l 0
= (g )q (r); (A4)
sm
k
m=1
l;s
where is the complex conjugate transp ose of the sth basis vector of irrep l , that is
k
pro jected out of an arbitrary function U of the Cartesian co ordinates, as describ ed in Eq.2
k
of the article. The symb ols in Eq. A4 indicate the inner pro duct b etween twovectors.
l;s
The identity in the second row follows from the prop erties of the basis functions :
k
1
symmetry op erations will only "mix up" co ordinates within a particular irrep. The equality
l;m
in the third row follows from the de nition of symmetry co ordinates q (Eq.4 of article).
k
By substitution from Eq. A4, Eq. A3 can now b e rewritten as:
X
1
l s; l;m
l 0 0 ;
0
)= U (r (g ) (g ) q (r)q (r) (A5)
g
sm
k; k
2
k l sm;
The expression in Eq. A5 can b e simpli ed considerably by summing it over all symmetry
0 1{3
op erations g and using the orthogonality prop erties of the irrep matrices. Thus, we
obtain:
X
1 1
l s; l;m
;
q (r)q (r) ; (A6) U (r)=
s m
l
k; k
2 s
p
k l sm;
where is the Kronecker delta function and s is the dimension of the irrep typ e T
ij p p
P
ls;s
1
l
corresp onding to l . Let = and substitute in Eq. A6 to obtain:
s
k;
k;
s
p
X X X X
1
l;m
l ;m
U = q (r)q (r); (A7)
k;
k
2
p m
l;2T k;
p
where wehave split up the summation over irreps l and into the pro duct of summations
over the irrep typ es T and the sum over l and that are of typ e T . Thus, b oth l and
p p
run from 1 to m , where m is the number of times irrep p o ccurs in the fully reduced
p p 2
representation (see Metho ds). From Eq. A7, we note the following: i) U is a sum over irrep
P
typ es T (U = u ), where each term
p p
p
X X X
l;m
l ;m
u = q q (A8)
p
k;
k
m
l;2T k;
p
dep ends only on symmetry co ordinates corresp onding to the irrep typ e T ; and ii) all terms
p
u split up into s terms (corresp onding to the di erentvalues of m):
p p
X X X
l;m
l ;m
u = u = q (r)q (r); (A9)
p pm
k;
k
m
l;2p k;
l
with identical co ecients , i.e., the co ecients are indep endentof m.
k;
In summary, the initial summation for U , shown in Eq. A1, is reduced to a series of
summations, where each of the terms corresp onds to one irrep typ e T . This corresp onds
p
to a blo ck diagonal form of the Hessian matrix in symmetry co ordinates. The size of the
individual blo cks is determined by m , as shown in Eq. A9 (indices l and run from 1 to
p
m ). For every irrep typ e T that o ccurs in the fully reduced representation, a diagonal
p p
blo ck of size m N m N app ears in . For irreps of dimension s > 1, the diagonal blo ck
p p p
l
of size m N m N is rep eated s times with identical co ecients ,andthus needs to
p p p
k;
b e calculated only once. 3
References
1. E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular vibrations. The theory of
infrared and Raman vibrational spectra (McGraw-Hill, New York, 1955).
2. G. Y. Lyubarskii, The application of group theory in physics (Pergamon Press, New York,
1960).
3. F. A. Cotton, Chemical applications of group theory, 3rd ed. (Wiley, New York, 1990). 4