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Determination of molecular vibrational force constants from kinematically defined normal coordinates
JOURNAL
OF MOLECULAR
SPECTROSCOPY
27,
27-13 (I!#%)
Determination of Molecular Vibrational from Kinematically Defined Normal
AiT Force
Cambridge
Research Bedford,
Force Constants Coordinates
Laboratories (O.lR) 1,. G. Hanncona Field, Massachusetts 01731
The criterion that the sum of the diagonal force cot&ants of a generalized valence force field attain an extremum, consistent with elimination of the arbitrariness inherent in t)he usual unrestricted formulation of the inverse secular problem of determining .l’$n(n + 1) vibrational force constants from only 12 harmonic frequencies and the equilibrium configuration, leads to II! solutions for the force field. All such solutions are characterized by symmetric GF matrices, and are obtained in terms of the observed vibrational frequencies, the equilibrium configuration, and the scaling chosen for the internal symmetry coordinates. The particular two force fields involving, respectively, t,he smallest and greatest, extrema of the trace of the F matrix are specified, and they correspond to completely reversed assignments of the frequencies. Comparison of the above model with two others also based on kinematicall> defined normal coordinat’es, viz., the method of the characteristic set of valenrr coordinates and the method of progressive rigidity, reveals that the last possesses some advantages. Examples including H&1, NO2 , and GeF4 are discussed. INTltODUCTION
For a complete non-redundant set of internal molecular vibrational symmetr!. coordinates, the general solution for the j in( II + 1) dist,inct element,s of the symmetric force constant mat)rix F associated \\-ith the n coordinates of a particular symmetry species (if it, is nowdegenerate, or wit,h a single component of t,he symmetry species if it is degener:ke) is given ( I-8 ) by t,he formula F =
ur-li2xA~r-1:4~,
I 11
The orthogonal matrix U and the diagonal matrix r are composed, respectively. of the eigenvectors and eigenvalues of Wilson’s (9) symmetric inverse kinetic, energy matrix G, which is diagonalized according to the scheme G = Ur8.
( L’‘I
The matrix A is diagonal, being formed from the harmonic freyuenciw vr (1. = 1, ..’ , n) in accordance wit#h the relation A, = 4g’~,‘. The ‘n X n matrix 27
25
FREEMAN
X is orthogonal (and chosen to have det X = + 1) ; but X is otherwise arbitrary, and is therefore a function of $in(n - 1) independent parameters. The infinite multiplicity of solutions for the $$z(n + 1) elements of F for a particular assignment of the n frequencies (corresponding to any one of then! possible permutations of the n diagonal elements of A along its leading diagonal) is expressed parametrically in Eq. (1) in terms of the matrix X. This inverse secular-eigenvalue problem is, as is well knows (l-6, IO, 11) , grossly underdetermined. The central difficulty is that, whilst t#he n eigenvalues A, of GF are completely defined by the n vibrational frequencies, the n eigenvectors comprising the transformation L from normal coordinates Q to symmetry coordinates S = LQ are given by L = Ur”‘X and are therefore indeterminate in the purely vibrational problem. The result is that infinitely many sets of >$n(n + 1) force constants, Eq. (l), and a corresponding infinity of normal coordinate transformations exist which exactly reproduce the n frequencies only. Attempts to render the vibrational problem fully determinate have, over the years, utilized two rather different approaches, sometimes in combination. The first approach, and the only physically really satisfactory method, involves the insertion into the problem of additional physical data on which the force field or normal coordinates depend. Such data include isotopic frequencies, Coriolis coefficients, centrifugal distortion coefficienk, and mean vibrational amplitudes. In a few favorable cases (usually small, light, very symmetrical molecules), a unique, or, at least, a highly preferred solution becomes possible for the general harmonic force field (12-18). However, for moderately complicated molecules, the only such additional data generally available are the isotopic frequencies, and, although the fitting of the parent and isotopic frequencies t#othe same harmonic force field does implicitly inject some independent information on vibrational amplitudes into the problem, it will not usually be sufficient to procure a unique solution for the ?&(n + 1) force constants. The most useful isotopic frequency shifts are those large shifts associated with the substitution of deuterium for hydrogen; but, unfort’unately it is precisely for these that the generally inaccurately known anharmonicities are also the largest. The possibility of utilizing smaller isotopic shifts has been investigated (19, 20)) but it seems that for small molecules the inclusion of Coriolis coefficients is, where possible, more effective in delimiting the possible force fields (12, 14). The other approach towards solving the vibrational problem involves the use of a restrictive model for the force field or for the normal coordinate transformation. In the former case, of which the simple valence, central, Urey-Bradley, and orbital-following force fields are widely used examples (9), the number of independent parameters in the force field is considerably less than the number of frequencies to be fitted, the interaction constants are defined in a systematic way, and the force constants therefore possess a degree of statistical and physical significance. It is noteworthy that, for an nth degree secular equation, the criteria for the existence of a purely diagonal valence force field are unknown (21-23).
VIBHATIONAI~
FORCE
CONSTANTS
“9
Only in occasional investigations, e.g., (20), are the constraint,s inherent in the definition of a model force field examined in any depth. For certain classes of molecules, notable successes have been achieved wi-ith model force fields, but such models must, to some extent, be regarded as empirical. So clear a p~~io~i preference obtains between tn-o such models t,ha,t fit identical data, and their relat,ive ability to accommodate further data is t,he deciding factor. E’or a secular equat,ion of the second degree, all possible solutions for the gener:d harmonic force field may bc plotSted as a function of just one independent varinbl(, ( 13-18, 24). For all secular equations of higher degree, there are too many inter:&on constants for a strictly analogous treatment to be practicable for thcl general harmonic force field, though a related technique ut,ilizing an analoguc computer has been described (11). The usual procedure for dealing nit,11 high order secular equations is the iterative refinement’, by one of the n-ell-kno\\n lc:lst squares techniques, of an initial approximation to a model force field. The COW vergence and other mathematical difficulties of these it,erative methods have hew discussed ( 12, 25-7), and it’ is emphasized that a refinement technique does not, itself contribute towards resolving the problem of the exist,ence of mult,iple solutions for t’he force constants. One difficultJy common to all such refinement t’echniques is the choice of an initial approximate force field. Oft’en t’his is merely an educated guess based upon considerations of t,he transferability of force constants (28), or upon difficultly assessable mathematical models in which how11, nonzero, off-diagorlnl G matrix elcment,s are ignored (29)) so that the full consequences implied in the initial approximation are not easy to discern. In this connection, various SJ.Stemat,ic models in which the normal coordinate t,ransformation is defined iI1 purely kinematic terms Cl-5, 7, 8, JO-3,$) can be useful, since t,hese models :\I‘(’ analyzable and lead to sets of ,lz~~(n + 1) readily calculable and always rcA:tl force constants that exactly reproduce the ‘~1harmonic frequencies. 111t,he prwent paper, one such method recently proposed (I, 30, 31) is idelltified with :I physical interpret’ation, analyzed in detail, and compared, in principle and by means of examples, to related methods. PROCEDURE It is recognized that diagonal force constants are generally more import,allt t,han ofl-diagonal force const,ants. In the procedure to be described for supplying the additional >&(n - 1) relations requisite to the unique determination of F from G and A, the physical interpretation imposed is t’he attainment bt_ the sum of the diagonal force constants of the greatest stationary value permitted by elimination of the arbitrariness usually implicit in the determinationof $$n(r~ + I) force constants from only n vibrational frequencies; that is, the solution sought corresponds to the great’est extremal sum of t,he diagonal force constants COT)sistent with all possible variations in the orthogonal ma,trix X in Eq. ( 1). Since the sum, c: F,, , of the diagonal force constants, i.e., t,he trace of t,hc
30
FREEMAN
F matrix, is invariant under a similarity transformation, (1) that
it follows from Eq.
= Tr ( r-11’2X~_fr-1’2)
The >$n(n + 1) independent orthogonality written as $
(X.,X,,
-
(3 1
relations for the matrix X may be
s,,j = 0
(4)
with 6,L = 1 for s = t, and 6,, = 0 for s f t. Solving the problem posed consists in finding the condition for Tr F, as given by Eq. (3)l, to be stationary with respect, to variations in the n2 elements X,, , subject to the $&(n + 1j equations of constraint, Eq. (4). The technique of I,agrangian undetermined multipliers is apposite : the necessary condition for constrained extrema of Tr F of Eq. (3) is identical to the necessary condition for unconstrained extrema of a related function H that is physically equivalent to Tr F and is defined as
in which the csl are the as yet undetermined multipliers, among which the relation E.?I= CfS
(6)
obtains in virtue of Eq. (4) because there are only $$n(n + 1) distinct multipliers. The condition necessary for const,rained extrema of Tr F is (7) so that
Equation (8) represents a set of n2 equations, which, together with the additional ?&(n + 1) equations of constraint, Eq. (4), serve to determine a total of n2 unknown elements X,, , together with the y&(n + 1) multipliers ~~~. 1The analogue of Ey. (3) for Tr(GF), of x.
Tr(GF)
=
za
A,,
is, of course,
independent
VIBRATIONAL
The 11’ eqwtions
comprising
FORCE
CONSTANTS
Eq. (8) may be expressed
r-‘XA + xc =
&=
81 in matrix
form: i $11
0
( 10)
-.W’XA.
from Eq. (ci), E is a symmet’ric matrix, E = E to Eq. (10) leads to the result
Hut,
x&w’
and application
of the condition
= r-‘X&F,
(11)
HO that the diagonal matrix r-’ must commute with the matrix XA~. (‘OIIsequentI\-, XA.~? must be a diagonal matrix. But A is itself diagonal, its leading elements being its eigenvalues; these, being invariant under an ort,hogonal similarity transformation, must also be the eigenvalues of XA8. Hence X must be such that the leading diagonal of the diagonal kansformed mat’rix XAT is merely one of t)he ‘YL! permutut’ions of the leading diagonal of A. Therefore XAB
= A*
(121)
or XA
= A*X
(1%)
\vhcre the leading diagonal of A* is any one of the II! possible permutSations of t,he quantities Ar = 4=‘v,’ (1. = 1, . . . , n). It, follows from t*he expanded form of Eq. ( 12b) that t’he orthogonal matrix X must be such that each of its rows COIItains only :L single nonzero element which must, be fl. The total number of di&nct orthogonal matrices X satisfying Eq. ( 12) is t’herefore Yn!, of which onl~ F’n ! correspond to det X = + 1. The facbor 2”-’ arises from the sign possibilit,iw for the
nonzero
correspond normal
elements
to normal
coordinate
of X, and coordinates
is immaterial,
matrices t,hat
there
differ
X t’hat differ
only
only
Since
in sign.
are only YIL!essentially
in this
respect,
t’he sign
different,
of ;I
solutions’
is X = E, the unit, matrix ) t,hat follow from of Tr F, Eq. (7). Uoreover, the same scat of II ! values of X will result from any one of the ‘n ! possible assignments of frequencies t#o internal symmetry coordinates. Consequently, the necessary conditions, Eq. (7) for stationary traces requires, from Eq. (I), that the forcr for X (of rvhich the most obvious
the
necessary
conditions
for extrema
” For example, in the 3 X 3 case, the B possible permlltations A* of the elements are generated from Eq. (1Za) by the following 6 essentially different X matrices: [%:;I,
[a_CJ
[-:i;],
[K],
[_;::I,
of L\
[HK].
The qllalification “essentially different” is needed becawe still other X matrices satisfying E(l. Wa) are possible. Each of the above 6 matrices, if subjected to sign changes in alI> pair rlf rows, gives rise to 3 additional tilln A* already obtained.
matrices which, however, merely repeat the permilt a-
FREEMAN
32
constant matrices be given by3 (13) and the corresponding traces by Tr F* = 2 I’;‘&*. r
(14)
The equivalence of the present results, Eq. (12) and Eq. (13)) to the requirement that the matrix GF be symmetric and given by GF = UA”o
(15)
is demonstrated by substituting Eq. (1) and Eq. (2) into the relationship GF = FG which specifies the symmetric nature of GF. This equivalence constitutes the identification of Billes’ recent method (1, SO, 31) for determining force constants with the present interpretation in terms of extremal values of summed diagonal force constants. Let us assume, without any loss of generality, that the rows and columns of G (and F) were ordered so as to cause the emergence of the eigenvalues I?,. of G in a monotonically decreasing sequence4 along the leading diagonal of I? in Eq. (2). 3The force field F* with a stationary trace can be diagonal, i.e., can be a simple valence force field (say Fd*), only if, from Eqs. (2) and (13), Fd*G = GFd*; i.e., only if G happens to be diagonal. Except in this trivial case, the present criterion, Eq. (7), does not permit useful general comment as to the relative magnitudes of the diagonal and off-diagonal force constants satisfying Eq. (7). * Degeneracy, as a consequence of molecular symmetry, among the eigenvalues of G is not troublesome since only a single component of a symmetrically degenerate species is to be considered in setting up Eq. (1). For the sake of Eq. (18), accidental degeneracy may be assumed absent because it is removable by re-scaling (3) the internal symmetry coordinates. In this connection it is emphasized that any solution for F obtained from Eq. (1) by setting the orthogonal matrix X equal to a special matrix depends intimately on the eigenvalues and eigenvectors of G, and hence not only on the equilibrium configuration (intramolecular geometry and masses) but also, to some extent, on the choice of basis coordinates for G. For the same X, physically different force fields are obtained corresponding to two differently scaled sets of internal symmetry coordinates. Thus, if primes refer to quantities based on a set of internal symmetry coordinates S’scaled differently from the set S, so that S’ = bS where d is a diagonal matrix of scaling factors, then F’ = 6-‘Fd-‘, G’ = dGd, and L’ = 7J’PizX’. It is obligatory to choose X’ = r’-1’2~‘~UP’2X in order to obtain physically identical force fields by the consecutive application of Eq. (1) to the two coordinate syst,ems. For example, the choice of X = E leads, from Eq. (4)) to a symmetric GF matrix (1) and to a stationary value of Tr F; similarly, the choice of X’ = E leads to a symmetric G’F’ matrix and to a stationary value of Tr F’; but the force fields F and F’ so obtained are not physically identical. To ensure the physical identity of the two force fields (i.e., to ensure that F’ = OFd-I), it is mandatory, if X is set equal to E, to set X’ equal to rr-l'z~dUrl'z. If, on the other hand, the two coordinate systems are orthogonally related
VIBRATIONAL
FORCE
CONSTANT8
:3:3
Then it is algebraically obvious that the greatest value5 of Tr F consistent with Eq. (7) is, from Eq. (14), Tr F. = 2
I’:‘&. ,
t 161
t,he corresponding force field being Fo =
Ur-‘~8
r1 > I-2 > . . . >
i 17) r,
(1Stt)
and A* < 1~2 <
.
< A,,
( 1Sb
)
in both Eq. (16 ) and Eq. (17), so t,hat the eigenvalue products in those equations correspond to the pairing pattern obtained with both eigenvalue sequences :,I’ranged monot80nically, but in opposite order. For this particular assignment of the frequencies to the internal symmeky co-ordinates, the setting of
X=E
Cl!))
E = -r-in
( “0 1
and
defines the greatest value of Tr F obeying the newwar)- conditions, Eq. (7 1, for an extremum. If this solution, Eqs. (16-20)) also fulfills the sufficient condition for an ext’rcmum of Tr F, then Eqs. (16) and (IS) give the greatest maximal sum of t,he diagonal force co&ants corresponding to the original choice of int,crn;l] symmetry coordinates. (and therefore identically scaled), the physical equivalence of the t’wo associated forc:e fields requires that X’ be set equal to X. Thus the extremum properties of the trace of the force constant matrix and the symmetric nature of the GF matrix are independent of t,ho choice among orthogonally related systems of internal symmetry coordinates. 5 It is clear that the smallest value of TrF consistent with Eq. (7) is obtained if the frerlllency assignment shown in Eq. (18) is completely reversed, so that r1 > rs >
> r,,
and A, > A2 >
> ‘La
The subsequent argument can, mutatis mzctandis, be based upon this premiss. Actual trials are needed to indicate whether the prescription of minimal, or of maximal, values of TrF leads to previously established frequency assignments and to physically acceptable values of force constants.
FKEEMAN
34
Admittedly, the present application of the necessary condition, Eq. (7)) for extrema of Tr F could formally be made even when such extrema do not exist, but the physical interpretation would then be undermined somewhat. It is therefore of interest to inquire briefly into the nature of the sufficient condition that a solution to Eq. (7) should indeed represent an extremal value. The condition for sufhciency is quite complicated (%), being dependent upon whether a certain quadratic form, generated from a Taylor series expansion about the solution to Eq. (7)) is positive definite (existence of a minimum) or negative definite (existence of a maximum) when subject to a set of linear equations derived from the equations of constraint, Eq. (4). The resultant condition for sufficiency is expressible in terms of the alternation or continuation of sign in the expanded power series form of a related secular determinant; but the analytic application of this criterion of sufficiency is too cumbersome to be worthwhile when more than very few frequencies are involved. It has been verified that, for two frequencies, Eqs. (1618) lead to the existence of a maximal trace and that the reverse frequency assignment leads to a minimal trace,5 so that the following is always true: (“1) When more frequencies are involved, empirical computer tests for sufficiency are probably adequate. The results obtained so far, concerning force fields with stationary traces, pertain directly to any one symmetry species of a molecule and, as a corollary, to a complete molecule that has no elements of symmetry. For a molecule with some symmetry the number of distinct force constants is reduced, and the trace of the matrix j of generalized valence force constants (based on N independent internal coordinates) is Tr j = Es faa , in which some of the fuarare identical and must be included multiply in the summation. This trace is the same as that of the symmetrized force constant matrix 5 (consisting of step-matrices F along its principal diagonal) since the symmetry-factoring within 5 is accomplished by subjecting j to an orthogonal similarity transformation. Furthermore, the necessary condition for extrema of Tr j is the same as that for extrema of the traces of all the individual step-matrices F comprising 5. In other words, the conclusions previously reached for a single symmetry species may be extended to any complete molecule, provided that the multiplicity of identical force constants is correctly included in the summation of the diagonal force constants. COMPARISON
WITH
RELATED
METHODS
For the present particular solution represented by Eqs. (16-20) formation L in the relationship S = LQ
the trans(22)
between internal symmetry coordinates S and normal coordinates Q is given by L = ur”2
(231
\‘IBHATIONAL
FORCE
CONSTANTS
3.5
Hince the general formula ( 1) for L, corresponding to Eq. ( 1) , is6 L = ur”?X.
(“4)
Equation (23) k a purely kinematic defir&ion of the normal coordinate tjrans;formation in t’erms of G matrix properties. Another such definition results from the choice (%5’) of x = 0:.
i “;i 1
and corresponds to the “method of the charact,eristic set of valence coordinates”, which has been shown (7,8), by means similar to the present, to satisfy the COII,dition necessary for a stat,ionary value of c: L,, , each L,, being defined positively. This solution (assumed sufficient for a maximum) is claimed (7, 8 I to yield t,he “most diagonal” form of L. It is easily seen, because 2 L:, = Tr (Le) T.S
= Tr G = $
I’, ,
t,hat the sum of the squares of all the elements of L is independent of the choiw of X; therefore the idea is intuitively plausible t’hat t*he choice of L ZZ ur”20,
( 27 1
which maximizes c: L,, , should lead to “small” off -diagonal L elements. Ho\\.ever, perhaps the most obvious way of interpreting the phrase-“most diagonal” form of L -- is to suppose that the sum of the squares of all the off-diagonal L LK be maximaL The condition elements, CY&. L”,8, be minimal, i.e., t’hat c: for this may, by methods similar to those used in Eqs. (3-ll), he shocvli to necessitate that X be chosen so that aA = A”x,
i”S)
n,hcre A is defined as t’he matrix wibh elements A,,* , A Tb55 /JS’ IT.SY P2. 7
(“9)
The choice of X = 0, Eq. (25), requires the mat’rix UA to be symmetric, which it is generally not; hence Eq. (27) does not lead t’o the “most diagonal” form of L, at least not. in the sense defined by a minimal value for c& LFs . It is also noteworthy that the condition, Eq. (25) or Eq. (27), that gives the greatest value of Tr L arising from the necessary condit8ion for exkema of Tr L does not itself differentiate among the n! possible frequency assignmentas st,ill available for the calculation of F from Eq. l 1). III addition to the two models already mentioned for the determination of force 6Because the sign of a normal coordinate is arbitrary, there are only n! essential11 different normal coordinate transformations L = lJ@X satisfying Ey. (12) and leading to stat,ionary values of TrF. See footnot,e 2. 7 It is worth noting that Lt, = ah,/aF,.,
FREEMAN
36
constants from kinematically defined normal coordinates, a third model called the method for progressive rigidity (.c?,%-8’4) is currently in use. A summarized comparison of these three models is presented in Table I. Only the row titled “Scaling invariance” requires further comment. The effect for model I of resealing the symmetry coordinates has been discussed previously in detaiJ4 and similar considerations obtain for model II. For any harmonic force field, the physical invariance of the potential energy, kinetic energy and normal coordinates to a scaling change from symmetry coordinates S to S’ = dS (d being diagonal) demands that F’ = ddlFddl, G’ = dGd, and L’ = dL; so that, although F and G individually undergo diagonal congruence transformations (related inversely to each other), the transformation of the product GF to G’F’ = dGFd? proceeds via a diagonal similarity transformation (albeit non-orthogonal), and eigenvalues A (but not r) are left unaltered, as are the normal coordinates Q, since Q’ = L’-‘S’ = L-1d-1dS = Q. However, the symmetric property of neither GF (model I) nor L (model II) is preserved under the scaling change, in contrast to the lower triangular property of L (model III) which is preserved. Of the three models, only the method of progressive rigidity is completely independent of the scaling choice for the symmetry coordinates, and is, in this respect, theoretically preferable to models I or II. In the important special case of a second degree secular equation, U and X may TABLE
I
PROPERTIES OF FORCE FIELDS WITH KINEMATICALLY DEFINED NORMAL COORDINATES Model I
Model II L symmetric
GF symmetric L = ur”2
Transformationa Force field
F = Vr-‘Ati
L = us@7 F = Vr-“2i?1hVr-“2~
Other relations
X=E
x=8
GF = FG EL = r
GL = LG EL = G
Model III Progressive rigidity
L,, = F,, = T, s (L-‘),, (GF),,
0, r < s c:: (L-‘),,(L-‘),& = 1, . . . ,n. = 0, r < s = 0, T < s
(GO, = A, G,,= ~LmLso,r 5 s
Extremal
property
Scaling invarianceb References * The transformation
LQ. b The
TrF stationary
TrL stationary
No
No
Each F,, stationary in the (partial) system comprising only the highest T modes. Yes
(1, 4, 90,n)
(I-J, 7, 8)
(SZ-34, 40-43)
between symmetry
coordinates
S and normal coordinates
Q is S =
refers to the physical invariance or entry corresponding to “scaling invariance” otherwise of the normal coordinates, force field, and extremal property under a change in scale of the symmetry coordinates. See the text.
VIBRATIONAL
conveniently
FORCE
CONSTANTS
37
be defined by the relations
so that the second degree equivalents spect,ively , be given as e=o
of Eqs.
( 19),
(sj,
and (28)
may, re-
(maximal value of F11 + Fyi),
0 = -C#J (maximal value of Ll1 + IJ?z) (
(31) (3”)
and
tan 2e= -
(I?1rz) l”’ l,.;(rl
+ r,>
tan 24, cos 30 cos 34 > 0( minimal value of L4, + Li,).
(:3:3)
The analogous expression for the model of progressive rigidity is tan0
= -
0
112
2
tan 4.
EXAMPLES
The empirical suitability of the three models is best tested by select,ing examples, such as HzO, SOZ, and GeF, , for which the general harmonic force fields can be established accurately. (a) m’atw. The water molecule is one of the few instances in which large isotopic frequency shifts are accurately known harmonic quantities. In Table II, each row of force constants for solutions I--XIII reproduces the frequencies of the molecule(s) shown at, the extreme left of the particular row, and t’he frequent! deviations are a measure of the failure of the force field to accommodate the isotopic data exactly. Solutions XIII, XIV, and XV are the best available, and the error estimates obtained from the present work and in Refs. (do,27 j suggest that Strey’s (SC) are optimistically small. Strey prefers solution XI but it is only marginally superior to several others, and the analogous solution XII for the isotopic molecule is among the least good of the series. Only solution IV is intolerably poor. A comparison of solution I tjo II and V 60 VI shows the effect, of resealing the angular symmetry coordinate by a factor chosen (as is cust,omrrr~ but, not necessary) equal to the bond length. Because this scale factor is close to unity, and for another reasona, the effect of such resealing on the force fields is not very marked. 8 It is easy to show that, if the symmetric L matrix (solution V) is nearly diagonal (as it might well be when TrL is maximal), then a scale change will produce a nearly symmetric L’ and nearly maximal TrL’. Similarly, if in addition to being symmetric, the matrix GF is nearly diagonal, then a scale change will produce a nearly symmetric G’F’ matrix and nearly extremal TrF’. Under these conditions, the effects on the force field of a scale change will be least pronounced.
TABLE
II
COMPARISON OF KINEMATICALLY DEFINED AND ITERATIVELY REFINED FORCE FIELDS FOR WATER AND ITS ISOTOPES
Solution
Characterization of force field
fr I II III IV \ l-1 VII VIII IX X XI XII XIII XIV XV
H,O: (TrF)z,d, H20 : (TrF’):,, D,O: (TrF):,, H20: (TrF)&,, Hz0 : (TrZ)“,,, H@: (TrZ’)&,, H,O: (Z;, + L:,)% DzO : (TrZ)kaX I-I,0 : progressive rigidity8 D20: progressive rigidityg H,O: D*O: Hz0 I-I,0 1120
8.448 8.445 8.447 5.058 8.452 8.452 8.453 8.463 8.444 8.429 8.454 8.468 8.453 8.455 8.451
(F,l)!:,,, (Fll)L + 1120: exact solution’ + DrO: iteration; + DzO + HDO: iteration”
* In terms of internal coordinates 2 x P.E.
= f,.(+
Frequency deviationb (cm-‘)
Force constants” (mdyn A-l)
(TV ,Q
r?fm
f7?
0.785 0.791 0.868 4.020 0.754 0.754 0.752 0.769 0.750 0.755 0.761 0.792 0.760 0.762 0.756
-0.107 -0.110 -0.108 -3.498 -0.103 -0.104 -0.103 -0.093 -0.111 -0.126 -0.100 -0.087 -0.102 -0.099 -0.104
rLi'fczMean
Max
1.9 2.1 6.1 109.3 1.3 1.3 1.7 1.6 2.2 2.7 0.8 2.9 0.7 0.8 1.2
6 7 27 534 4 4 5 5 8 8 5 13 5 5 3
0.388 0.414 0.701 -0.108 0.159 0.157 0.140 0.299 0.044 0.841 0.236 0.441 0.225 0.246 0.168
, CX)the potential energy is given by
+ T?) + faa2 + 2frr~l~ + 2f,,(~1 + T&Y,
in which the force constant units are defined as mdyn A+ for fV and frr , mdyn for fTa , and mdyn A for fe . The quantities listed in Table II are expressed in uniform units of mdyn A-1, with ~0 = 0.95728. The unprimed kinetic energy matrix, constructed from the equilibrium parameters ~0 = 0.95728 and (~0 = 104.52” @6), is defined and scaled as in Ref. (9). The symmetry coordinates used to form P and G matrices are &@I)
= 2-“%
+ TZ),
S&41)
= cy, and
&(B1)
= 2-1’2(rl -
rz),
so that
2 x P.E.
= (f, + f,,)SP
+ 2 21’2f,S1Sz
+
fc,S,z+ (fr - f&3,‘.
b The deviation is the magnitude of the wavenumber difference between the computed harmonic frequency and the harmonic frequency derived from experiments [and quoted in Ref. (,W,96)] ; 10 such accurately known frequencies, viz., WI , wx, and WQof each of l&O, DzO and HDO and ws of T20 are used. c The matrices F and Z here refer to only the two totally symmetric vibrations of the indicated molecule. However, extrema of Tr F = f, + fr, + fa and of 2f, + .L occur together. d If this sohltion were exactly transferable from Hz0 to DZO, the skew-symmetric com0.0524 0 mutator [Glr,o , Gn,ol would vanish (1) ; in fact, [GB%o , Go,01 = 0 1 -0.0524 A primed matrix is based on resealed symmetry coordinates S’ = ds with
1
0
d= [
10 oroo.
0
00
1
1
f See Equation (33) in the text. g In the present, case of a second degree secular equation, minimal (24). h In the present case of a second degree secular equation, and GF are upper triangular matrices (24). i This solution is obtained by elementary algebra. j See Table III of Ref. (36). k See Ref. (.t?‘?‘) and Table VIII of Ref. (20). 38
this model implies that FB is this condition
implies that Z
VIBRATIONAL
FORCE TABLE
CONSTANTS
39
III
MODEL FORCE FIELDS FOR “NO* Characterization of force field
Solution
Force constants” (mdyn .-\-‘)
I__ I II III 11. 1. \‘I \I1 i-111
14N02:
(TrF)kx
14N0z: (TrF’);,, ‘%Or : (TrL) !kx 14N0t : (TrL’);:x 14N0z: (L”12+ &&xi,, 14N02: progressive rigidityd” Wo?:
(F&:x
%Or,
15NOn:
il Footnote 1)Footnote c Footnote <’ See Ref. 0 See Eq. i Footnote g Foot,notc ‘8 See I
a of Table c of Table e of Table (96). (33) in the g of Table h of Table (37).
II3 + microwave’*
12.086 12.156 12.084 12.117 12.214 11.306 12.217 / 11.033 l\*o.o5
II applies, except, that ru = l.l%iA II applies. II applies.
1.830 1.725 1.283 1.306 1.102 1.107 1.525 1.109 10.01
3.188 3.258 3.187 3.216 3.316 2.405 3.31ti 2.110 zto.05
2.634 2.483 1 633 1. CL5 1.184 0.m
2.170 0.1x1 zlA~.011
and u,, = 131”1’ (8’7 I_
test. II applies. II applies.
(b) Nitqen dioxicle. The best available force field (solut,ion VIII, Table III), incorporating microwave and infrared frequency dat,a (3’) for both of tl:r isotopic species ‘*SO2 and '%Or , is compared in Table III to the various kinematic models. The progressive rigidity model (solution VI) is clearly supcriol to t,he others which cope rather poorly with the interaction const,ants. The effects on the calculated force fields of resealing the symmetry coordinatjes are seen b! comparing solution I to II, and III to lV, to be more pronounced than for the wattrr molecule. For ‘“NO:! t’he difference bet’ween solution III, for nhich ( Lll + L,,) is maximal, and solut8ion V, for which L is “most diagonal” in the sense that (I,:, + Li1) is minimal [and (Lil + L&J maximal]. is more marked than for HZ0 (compare solutions V and VII of Table II). (c) Gew~anz’uw tet~q’ka,*ide. The general yuadrat’ic force field for t)hr !“! species is rather sharply defined by the requirement t,hat the force constants tii not, only the frequencies but also the Coriolis coefficients determined from the esperimental P-R spacings in the infrared spectrum ( 16). The value of l4 shown in Fig. 3 of the paper by J,evin and Abramo\vitz ( 16) differs from its correct valur which is given in Table I of t,hat study. The corrected values of t)he symmetr) force const,ants are those in Table IV (solution VII) of the present workY, and not those in their Table IV. Of the various model force fields in the present 9 These corrected valnes from Dr. I. W. Leviu.
were supplied
in a private
comrnllnicatiou
(Jalluary
191%)
40
FREEMAN TABLE
IV
MODEL FORCE FIELDS FOR GEF~ (Fz SPECIES) Solution
Characterization of force field
I II III IV V VI
(TrL’)mex (TrL):,, Progressive (F’a&ex
VII
Exp. Coriolis
Force constants” (mdyn A-l) FIR
(TrF’)max
6.102
(TrF)“,ax
3.987
rigidity
6.176 6.105 5.671 6.266 5.87 f0.05
& -1.813 -2.455 -0.848 -0.699 -0.160 -1.288 -0.32 f0.05
F’46 0.787 1.892 0.362 0.328 0.271 0.507 0.27 10.01
Coriolis coefhcientsb
r3a -0.38 -0.42 -0.04 +o.oa f0.26 -0.24 +O.ZOd
.c44 i-O.88 +0.92 f0.54 $0.48 +0.24 +0.74 i-0.354
&The force constant and kinetic energy matrices are based on symmetry coordinates Sa’ = ,14(ri - T%- r1 + ~4) and Sa’ = r02-‘@ (01~~ - o[i4) defined in terms of internal coordinates ri and ~lij and,th,ez equilibrium paramet:rs (16) TO = 1.67A, 010 = 109”28’. The potential energy is M(Fd% + 2F&‘S4’ + F&I 1 in which the force constant units are mdyn A-‘. b These are calculated from the following formulae, which are essentially those of Ref. (58), but with some signs changed ty confyrm to t$e usage of Ref. (16) : (33+ ska= 1 and r33 = (A, - A&i [Al + 5s 114- pp(Fz3 f Fad - 2F34)1, in which /*p is the reciprocal of the atomic mass of fluorine. c An unprimed matrix is based on the resealed symmetry coordinates SO = S3’ and sq = r,’ Sa’. d The experimental Coriolis coefficients obey the sum rule only approximately. See Ref. (16) and the text.
Table IV, the model of progressive rigidit’y gives the most satisfactory results, especially as regards the interaction constant and the Coriolis coefficients”. The marked differences between solutions I and II, and, to a lesser extent, between solutions III and IV show the dependence of these kinematic models on the scaling of the symmetry coordinates. (d) Othey molecules. The model with symmetric GF and extremal Tr F (Table I, model I) might be expected, from Tables III and V of Ref. (SC), to yield fairly satisfactory force constants for H&S (and D2S) and HzSe (and D2Se) but not for SO, or C102 . Plausible results are obtained for C&H, (30) and, it is stated (31)) for the Al, species of CeH6 ; but in neither case were deuterium substituted molecules studied. Model I is claimed (31) to result in unconventional force fields for RF,, BCl, , and SF,. lo For SiF4 , the force fields of Duncan and Mills (14) and of Shimanouchi et al. (20) are based on an estimated value of 53 = 0.63 for which no great reliability was claimed (14). The more recent determination by Levin and Abramowitz (16) gives 5‘3 = 0.49, which value corresponds to a force field in good agreement with the model of progressive rigidity. However, the observed value of rh = -0.12, which leads to a sum rule inconsistency, is still an unresolved difficulty.
VIBRATIONAL
FORCE
CONSTANTS
-41
The model with symmetric L and extremal Tr L (Table I, Model II) has been t#ested for the alkyl (3) and silyl (3, 4) halides; the results are reasonable compared to hybrid orbital force fields (12, 39)) although certain insen&ive intcraction constank constrained t’o remain zero in t’he hybrid orbital force firld att,ain values up to 0.S mdynA_’ in model II. The agreement bet#n-een the :I(‘curately known force constants (13) and model II (7, 8) is good for SiH, and fair for SiD, . Model II yields force constant,s that are reasonable for H,S C:uld D?S) and HSe (and D?Se) but poor for SO2 and Cl02 [Table V of Ref. ( .Sj ) I. Evaluation of the application to C?H&l (8) and HaBOa i A4) is dificult If it is remembered that the model of progressive rigidit,y and t)he :~tt:tinment of a minimal value for Fzf are equivalent’ for a second degree secular eq~l:~tion, :I variety of molecules becomes available for which this model may be evaluated crit,ically. The model is a good approximat’ion for the following: SO? (.I/; i ; H,S (.%);H,Se (36); CHJ, CDJ, CT4 (13);XH4, XD,(X = Si, Ge) (13);l~,‘spc~+~s of XHs , XD,(X = S, I’, As, Sb) (13); E species of AsI::{ ( 16) ; F1,, species of XFs(X = Se, Te, W) (17, 28‘); GeCl, (,/to, .(l 1. The model is a fair approsim:ltion for the following: ClO, (36); E species of BX,(X = F, Cl, Br, I) (1~5); E sp(xcies of PI?, (16‘); CX,(X = P‘, Cl, Br) (I/,, II?, ,$O, ,il ). The model is a poor :Lpprosimation for the following: A1 speicies of XH, , XD,( X = S, I’, t\s, Sb) ( I,Z) ; I:’ species of NF, (16) ; Fl,, species of SIC0 (17). In all of the above C:LSESI hc force fields are rather narrowly const)rained by experimentally known (‘oriolis coefficients, centrifugal distortion coefficients, or mean square vibraLtional amplitudes. The model of progressive rigidity has also been applied to the follo\ving : I-X4(X = F, Cl, Br; I- = I’b, Zr, Hf) (42, @) ; A1 species of CH,C’l, (.I,” ) ; A-1 ‘, species of C2H4 (3%‘); CeH,, (33). In t’hese cases, ho\vever, t)hc true force ti~ltls :ire not so accurately known. (c) Conclusion. From the limited number of examples surveyed, the provisional conclusion is drawn that the model of progressive rigidit,?- is the most satisfactory of the t’hree kinematic models examined in this paper. It is noted. however, that most of the molecules t)o which crit’icxl t,ests can be applied possc~s symmetry species containing only a fen- well-spaced frequencies, and that this is likely to be a necessary, though not’ a sufficient 1condition for success of the moclr~l of progressive rigidit,v. SUMMA11 Y ( a) The criterion that the sum of the diagonal force constants of a gencrnlized valence force field fulfill the condition necessary for att#ainment, of a stutiollary value, consistent with elimination of the arbitrariness inherent in the usual unrestricted formulation of t’he inverse secular problem of determining ~.$L(?L + 1) svmmetry force constants from t’he equilibrium geometry andonlJ !L vibrational frequencies, leads to ‘n! solutions for the force field, E(l. (13). Thr force fields so determined are independent of the choice of orthogonall\related
FREEMAN
42
coordinate systems but are dependent on the scaling chosen for the internal symmetry coordinates. These n! solutions, previously discussed in another context ( 30,31) are further characterized (1) by the requirement that their GF matrices be symmetric. The circumstances under which these solutions are transferable to an isot,opic molecule have also been examined (I). (b) A narrowing of the scope of the above criterion to the condition necessary for the greatest extremum of the trace of the F matrix leads to a unique solution for the force field, Eq. (17). The appropriate normal coordinate transformation, Eq. (23), is specified in terms of the eigenvalues and eigenvectors of the G matrix. The corresponding assignment of the vibrational frequencies to a sequence of the chosen internal symmetry coordinates is determined by the sequence of eigenvalues of the G matrix, in accordance with Eq. (18). The reverse of the vibrational assignment represented by Eq. (18) corresponds to the condition necessary for the smallest extremum of the trace of the F matrix. (c) The above conclusions concerning force fields with stationary traces relate specifically to a single symmetry species of a molecule. Their extension to any complete molecule is indicated and involves 110 novel features. (d) The analytic basis and physical interpretation of one method of determining molecular vibrational force constants from kinematically defined normal coordinates has been examined in detail and compared to two other such methods, viz., the method of the characteristic set of valence coordinates, and the method of progressive rigidity. The last method has two advantages: it alone is completely independent of the scaling of the internal symmetry coordinates, and, it appears, from an empirical examination of a limited number of examples, to be the most satisfactory of the t’hree kinematic models tested. Any of the three models yields real and readily calculable sets of force constant’s which may be of use in cases where Coriolis coefficients etc., are not available. RECEIVED:
October 25, 1967 REFERENCES
1. D. E. FREEMAN, J. Mol. Spectru. 22, 305 (1967). 2. P. PIJL~Y AND F. Tti~ii~, Acta Chim. Acad. Sci. Hung. 44, 287 (1965). 3. P. PULAY _~ND F. Tij~ii~,Acta Chim. Acad. Sci. Hung. 47, 273 (1966). 4. P. PULAY, Acta Chim. Acad. Sci. Hung. 62, 49 (1967). 5. F. Tij~ii~,Acta Chim. Acad. Sci. Hung. 62, 205 (1967). 6. S. TOMAN AND J. PLIVA, J. Mol. Spectry. 21, 362 (1966). 7. J. HERRaN AND F. CASTARO, Spectrochim. Acta 22, 1965 (1966). 8. J. HERRANZ BND F. CASTARO, Anales Real Sot. Espan. Fis. Quim. (Madrid) Ser. A 62, 199 (1966). 9. E. B. WILSON,J. C.DECIUS, AND P. C. CROSS, “Molecular Vibrations”. McGraw-Ilill, New York, 1955. IO. J. M. RUTH AND R. J. PHILIPPE, J. Chem. Phys. 41, 1492 (1964). 11. W. J. LEHMANN, L. BECKMBNN, AND L. GUTJAHR, J. Chem. Phys. 44.1654 (1966). Id. J. ALDOUS .IND I. M. MILLS, Spectrochim. Acta 18, 1073 (1962); 19, 1567 (1963).
VIRIUTIONAL
FORCE
CONSTANTS
1X
1s. J. I,. I)I.N(‘.iN :\NL)I. M. MILLS, Speclrochi~~~. dicta 26, 523 (196-1). MILLS, Speclrochinl. 11&a 20, 1089 (1964). 15. I. W. LKVIN AND R. AIX~MO~~ITZ, J. C’henl. Php. 43, 4213 (1965). 16. I. W. LIXIN Z\~~ S. ABRAMO\VITZ, J. C’hem. Ph!ys. 44, 2562 (196ti); A. I:~.oFF, Spec(/,ochim 2I cla 23A, 2421 (1967). 17. S. AHI(.\MO\VITZ .IND I. W. LEVIN, J. C’hem. Phys. 44, 3353 (1966). 18. S. A~cR.znlow~Tz .~ND I. W. LEVIR‘, rnorg. Chem. 6, 538 (1967). 19. M. Tsrrw~, J. dfol. Specflv. 19, 4 (1966). 20. ‘1’. SHIM.\N~UCHI, I. SIKAG.\T~.\, J. HIR.~IRHI, ANI) M. ISHII, J. 3101. Spec/r!/. 19, 78
14. J. I,. I)VNVz\K .~NU I. M.
(19ti6). 21. A. C. I)O\~XING .~NI) A. ti. HOUSEHOLDXR, J. .Issuc. fZ”ow~p. Uach. 3, 203 (1956). ,%?. J. IJHLIG, Z. anger. Math. Mech. 40, 123 (1960). d,?. Il. IIKINRICH, Z. angew. Math. Alech. 40, 62 (1960). 24. P. TORKINGTON, J. Chem. Phys. 17, 357 (1949). 25. 1~). E. FRIDXAN AND I. G. Ross, Spectrochint. aLctu 16, 1393 (1960). 26. M. J. BRUTON .\ND L. A. WO~D\~ULD, Spectrochim. ;lcta 23A, 175 (1967). $7. T. ~~HIM.~NOUCHI END I. SUZUKI, J. (‘hem. Phys. 42, 296 (1965); 43, 1854 11965). ,?a. D. E. FREEMAN .~NU M. K. Wrr~sos, Spectrochim. .lctu 21, 1825 (1965). 29. A. F.~DINI, Z. ~Vaturforsch. 21a, 426 11966); H. J. BECHER .\NU II. M~TTKS, Spectrochim. dctu 23A, 2449 (1967). SO. F. BILLES, ;ic(u Chim. _-lead. Sci. Hung. 47, 53 (1966). $1. S. J. CY\-IN, L. A. KRISTL\NSI:N, ANI) J. HHUNVOLI,, _-lcta C’h&l. .Icczr/. Sci. Hung. 51, 217 fIBfiT). ,!I$?. P. TOKKIXGTON, J. Chem. Phys. 17, 1026 (1949). 33. M. L.~Rx.\~DIE, J. Phys. Rd. 16, 365 (1954). $4. I. \‘. 0~1.ov.4 AND I. N. C+ODNEV, Opt. i Spektroskopiya 6, -147 (1959); Opt. SP~O.!/. (USSR1 6, 284 (1959J. of Maxima and Minima”. Uover, New York. 1960. 55. II. II.QYcocK, “Theory 56. (:. STni~;y, b. IwoZ. Specfry. 26, 87 (1967). 37. (;. I:. BII~D, J. C. B.\IHD, A. W. JACHII:, J. A. HODG~SON, le. F. C~HL, A. C. KUNKLI.,, ,I. iv.
HR.\SSFORD, J.
I<.4STRUP-ANDERSEN,
.~ND J. ROSENTH,~L,
(1964). $8. S. J. CYVIN, J. BRUNVOLL, .\ND B. N. CYVIN, Bull. $9. I. M. MILLS, Spectrochinr. Ada 19, 1585 (1963).
.I.
Chem.
Belg.
73,
Phjys.
40,
3378
40. V. h’.
~INOGRADOV~
Opt. Speclry.
41, V.
N. Phys.
J.,
No.
42, J. NV.C;ODNEV, 271,
4.9. A.
M.
844
(‘him.
,~ND I. N. ~;ODNXV, Opf. i Spektroskopiya,
(L’SSR),
VINOGRAUOV~
Sot.
8uppl.
AND I. N.
120 (1964).
Suppl. No. 2, 76 (1!)63);
No. 2, 40 (1966). Fizika,
No.
1, 57 (1965);
BND I. V. &GINA,
Opt.
i Spektroskopiyu
GODNJIV, Izvest.
r’L’%
Soviet
1, 47 (1965). A. M. ALEKs.~NDRovSsaY:~,
(1959); Opt.
Spectry.
ALEKS.~NDROV~KAYA
10, 27 (1961); Opt.
Spectry.
(USSR)
.~ND I. N. (USSR)
7, 172, 495
GODNEV, Opt.
7,
(1959). i Spektroskopiyu
9, 144 (1960); 10, 14 (1961).
9, 273
(196OJ;
OF MOLECULAR
SPECTROSCOPY
27,
27-13 (I!#%)
Determination of Molecular Vibrational from Kinematically Defined Normal
AiT Force
Cambridge
Research Bedford,
Force Constants Coordinates
Laboratories (O.lR) 1,. G. Hanncona Field, Massachusetts 01731
The criterion that the sum of the diagonal force cot&ants of a generalized valence force field attain an extremum, consistent with elimination of the arbitrariness inherent in t)he usual unrestricted formulation of the inverse secular problem of determining .l’$n(n + 1) vibrational force constants from only 12 harmonic frequencies and the equilibrium configuration, leads to II! solutions for the force field. All such solutions are characterized by symmetric GF matrices, and are obtained in terms of the observed vibrational frequencies, the equilibrium configuration, and the scaling chosen for the internal symmetry coordinates. The particular two force fields involving, respectively, t,he smallest and greatest, extrema of the trace of the F matrix are specified, and they correspond to completely reversed assignments of the frequencies. Comparison of the above model with two others also based on kinematicall> defined normal coordinat’es, viz., the method of the characteristic set of valenrr coordinates and the method of progressive rigidity, reveals that the last possesses some advantages. Examples including H&1, NO2 , and GeF4 are discussed. INTltODUCTION
For a complete non-redundant set of internal molecular vibrational symmetr!. coordinates, the general solution for the j in( II + 1) dist,inct element,s of the symmetric force constant mat)rix F associated \\-ith the n coordinates of a particular symmetry species (if it, is nowdegenerate, or wit,h a single component of t,he symmetry species if it is degener:ke) is given ( I-8 ) by t,he formula F =
ur-li2xA~r-1:4~,
I 11
The orthogonal matrix U and the diagonal matrix r are composed, respectively. of the eigenvectors and eigenvalues of Wilson’s (9) symmetric inverse kinetic, energy matrix G, which is diagonalized according to the scheme G = Ur8.
( L’‘I
The matrix A is diagonal, being formed from the harmonic freyuenciw vr (1. = 1, ..’ , n) in accordance wit#h the relation A, = 4g’~,‘. The ‘n X n matrix 27
25
FREEMAN
X is orthogonal (and chosen to have det X = + 1) ; but X is otherwise arbitrary, and is therefore a function of $in(n - 1) independent parameters. The infinite multiplicity of solutions for the $$z(n + 1) elements of F for a particular assignment of the n frequencies (corresponding to any one of then! possible permutations of the n diagonal elements of A along its leading diagonal) is expressed parametrically in Eq. (1) in terms of the matrix X. This inverse secular-eigenvalue problem is, as is well knows (l-6, IO, 11) , grossly underdetermined. The central difficulty is that, whilst t#he n eigenvalues A, of GF are completely defined by the n vibrational frequencies, the n eigenvectors comprising the transformation L from normal coordinates Q to symmetry coordinates S = LQ are given by L = Ur”‘X and are therefore indeterminate in the purely vibrational problem. The result is that infinitely many sets of >$n(n + 1) force constants, Eq. (l), and a corresponding infinity of normal coordinate transformations exist which exactly reproduce the n frequencies only. Attempts to render the vibrational problem fully determinate have, over the years, utilized two rather different approaches, sometimes in combination. The first approach, and the only physically really satisfactory method, involves the insertion into the problem of additional physical data on which the force field or normal coordinates depend. Such data include isotopic frequencies, Coriolis coefficients, centrifugal distortion coefficienk, and mean vibrational amplitudes. In a few favorable cases (usually small, light, very symmetrical molecules), a unique, or, at least, a highly preferred solution becomes possible for the general harmonic force field (12-18). However, for moderately complicated molecules, the only such additional data generally available are the isotopic frequencies, and, although the fitting of the parent and isotopic frequencies t#othe same harmonic force field does implicitly inject some independent information on vibrational amplitudes into the problem, it will not usually be sufficient to procure a unique solution for the ?&(n + 1) force constants. The most useful isotopic frequency shifts are those large shifts associated with the substitution of deuterium for hydrogen; but, unfort’unately it is precisely for these that the generally inaccurately known anharmonicities are also the largest. The possibility of utilizing smaller isotopic shifts has been investigated (19, 20)) but it seems that for small molecules the inclusion of Coriolis coefficients is, where possible, more effective in delimiting the possible force fields (12, 14). The other approach towards solving the vibrational problem involves the use of a restrictive model for the force field or for the normal coordinate transformation. In the former case, of which the simple valence, central, Urey-Bradley, and orbital-following force fields are widely used examples (9), the number of independent parameters in the force field is considerably less than the number of frequencies to be fitted, the interaction constants are defined in a systematic way, and the force constants therefore possess a degree of statistical and physical significance. It is noteworthy that, for an nth degree secular equation, the criteria for the existence of a purely diagonal valence force field are unknown (21-23).
VIBHATIONAI~
FORCE
CONSTANTS
“9
Only in occasional investigations, e.g., (20), are the constraint,s inherent in the definition of a model force field examined in any depth. For certain classes of molecules, notable successes have been achieved wi-ith model force fields, but such models must, to some extent, be regarded as empirical. So clear a p~~io~i preference obtains between tn-o such models t,ha,t fit identical data, and their relat,ive ability to accommodate further data is t,he deciding factor. E’or a secular equat,ion of the second degree, all possible solutions for the gener:d harmonic force field may bc plotSted as a function of just one independent varinbl(, ( 13-18, 24). For all secular equations of higher degree, there are too many inter:&on constants for a strictly analogous treatment to be practicable for thcl general harmonic force field, though a related technique ut,ilizing an analoguc computer has been described (11). The usual procedure for dealing nit,11 high order secular equations is the iterative refinement’, by one of the n-ell-kno\\n lc:lst squares techniques, of an initial approximation to a model force field. The COW vergence and other mathematical difficulties of these it,erative methods have hew discussed ( 12, 25-7), and it’ is emphasized that a refinement technique does not, itself contribute towards resolving the problem of the exist,ence of mult,iple solutions for t’he force constants. One difficultJy common to all such refinement t’echniques is the choice of an initial approximate force field. Oft’en t’his is merely an educated guess based upon considerations of t,he transferability of force constants (28), or upon difficultly assessable mathematical models in which how11, nonzero, off-diagorlnl G matrix elcment,s are ignored (29)) so that the full consequences implied in the initial approximation are not easy to discern. In this connection, various SJ.Stemat,ic models in which the normal coordinate t,ransformation is defined iI1 purely kinematic terms Cl-5, 7, 8, JO-3,$) can be useful, since t,hese models :\I‘(’ analyzable and lead to sets of ,lz~~(n + 1) readily calculable and always rcA:tl force constants that exactly reproduce the ‘~1harmonic frequencies. 111t,he prwent paper, one such method recently proposed (I, 30, 31) is idelltified with :I physical interpret’ation, analyzed in detail, and compared, in principle and by means of examples, to related methods. PROCEDURE It is recognized that diagonal force constants are generally more import,allt t,han ofl-diagonal force const,ants. In the procedure to be described for supplying the additional >&(n - 1) relations requisite to the unique determination of F from G and A, the physical interpretation imposed is t’he attainment bt_ the sum of the diagonal force constants of the greatest stationary value permitted by elimination of the arbitrariness usually implicit in the determinationof $$n(r~ + I) force constants from only n vibrational frequencies; that is, the solution sought corresponds to the great’est extremal sum of t,he diagonal force constants COT)sistent with all possible variations in the orthogonal ma,trix X in Eq. ( 1). Since the sum, c: F,, , of the diagonal force constants, i.e., t,he trace of t,hc
30
FREEMAN
F matrix, is invariant under a similarity transformation, (1) that
it follows from Eq.
= Tr ( r-11’2X~_fr-1’2)
The >$n(n + 1) independent orthogonality written as $
(X.,X,,
-
(3 1
relations for the matrix X may be
s,,j = 0
(4)
with 6,L = 1 for s = t, and 6,, = 0 for s f t. Solving the problem posed consists in finding the condition for Tr F, as given by Eq. (3)l, to be stationary with respect, to variations in the n2 elements X,, , subject to the $&(n + 1j equations of constraint, Eq. (4). The technique of I,agrangian undetermined multipliers is apposite : the necessary condition for constrained extrema of Tr F of Eq. (3) is identical to the necessary condition for unconstrained extrema of a related function H that is physically equivalent to Tr F and is defined as
in which the csl are the as yet undetermined multipliers, among which the relation E.?I= CfS
(6)
obtains in virtue of Eq. (4) because there are only $$n(n + 1) distinct multipliers. The condition necessary for const,rained extrema of Tr F is (7) so that
Equation (8) represents a set of n2 equations, which, together with the additional ?&(n + 1) equations of constraint, Eq. (4), serve to determine a total of n2 unknown elements X,, , together with the y&(n + 1) multipliers ~~~. 1The analogue of Ey. (3) for Tr(GF), of x.
Tr(GF)
=
za
A,,
is, of course,
independent
VIBRATIONAL
The 11’ eqwtions
comprising
FORCE
CONSTANTS
Eq. (8) may be expressed
r-‘XA + xc =
&=
81 in matrix
form: i $11
0
( 10)
-.W’XA.
from Eq. (ci), E is a symmet’ric matrix, E = E to Eq. (10) leads to the result
Hut,
x&w’
and application
of the condition
= r-‘X&F,
(11)
HO that the diagonal matrix r-’ must commute with the matrix XA~. (‘OIIsequentI\-, XA.~? must be a diagonal matrix. But A is itself diagonal, its leading elements being its eigenvalues; these, being invariant under an ort,hogonal similarity transformation, must also be the eigenvalues of XA8. Hence X must be such that the leading diagonal of the diagonal kansformed mat’rix XAT is merely one of t)he ‘YL! permutut’ions of the leading diagonal of A. Therefore XAB
= A*
(121)
or XA
= A*X
(1%)
\vhcre the leading diagonal of A* is any one of the II! possible permutSations of t,he quantities Ar = 4=‘v,’ (1. = 1, . . . , n). It, follows from t*he expanded form of Eq. ( 12b) that t’he orthogonal matrix X must be such that each of its rows COIItains only :L single nonzero element which must, be fl. The total number of di&nct orthogonal matrices X satisfying Eq. ( 12) is t’herefore Yn!, of which onl~ F’n ! correspond to det X = + 1. The facbor 2”-’ arises from the sign possibilit,iw for the
nonzero
correspond normal
elements
to normal
coordinate
of X, and coordinates
is immaterial,
matrices t,hat
there
differ
X t’hat differ
only
only
Since
in sign.
are only YIL!essentially
in this
respect,
t’he sign
different,
of ;I
solutions’
is X = E, the unit, matrix ) t,hat follow from of Tr F, Eq. (7). Uoreover, the same scat of II ! values of X will result from any one of the ‘n ! possible assignments of frequencies t#o internal symmetry coordinates. Consequently, the necessary conditions, Eq. (7) for stationary traces requires, from Eq. (I), that the forcr for X (of rvhich the most obvious
the
necessary
conditions
for extrema
” For example, in the 3 X 3 case, the B possible permlltations A* of the elements are generated from Eq. (1Za) by the following 6 essentially different X matrices: [%:;I,
[a_CJ
[-:i;],
[K],
[_;::I,
of L\
[HK].
The qllalification “essentially different” is needed becawe still other X matrices satisfying E(l. Wa) are possible. Each of the above 6 matrices, if subjected to sign changes in alI> pair rlf rows, gives rise to 3 additional tilln A* already obtained.
matrices which, however, merely repeat the permilt a-
FREEMAN
32
constant matrices be given by3 (13) and the corresponding traces by Tr F* = 2 I’;‘&*. r
(14)
The equivalence of the present results, Eq. (12) and Eq. (13)) to the requirement that the matrix GF be symmetric and given by GF = UA”o
(15)
is demonstrated by substituting Eq. (1) and Eq. (2) into the relationship GF = FG which specifies the symmetric nature of GF. This equivalence constitutes the identification of Billes’ recent method (1, SO, 31) for determining force constants with the present interpretation in terms of extremal values of summed diagonal force constants. Let us assume, without any loss of generality, that the rows and columns of G (and F) were ordered so as to cause the emergence of the eigenvalues I?,. of G in a monotonically decreasing sequence4 along the leading diagonal of I? in Eq. (2). 3The force field F* with a stationary trace can be diagonal, i.e., can be a simple valence force field (say Fd*), only if, from Eqs. (2) and (13), Fd*G = GFd*; i.e., only if G happens to be diagonal. Except in this trivial case, the present criterion, Eq. (7), does not permit useful general comment as to the relative magnitudes of the diagonal and off-diagonal force constants satisfying Eq. (7). * Degeneracy, as a consequence of molecular symmetry, among the eigenvalues of G is not troublesome since only a single component of a symmetrically degenerate species is to be considered in setting up Eq. (1). For the sake of Eq. (18), accidental degeneracy may be assumed absent because it is removable by re-scaling (3) the internal symmetry coordinates. In this connection it is emphasized that any solution for F obtained from Eq. (1) by setting the orthogonal matrix X equal to a special matrix depends intimately on the eigenvalues and eigenvectors of G, and hence not only on the equilibrium configuration (intramolecular geometry and masses) but also, to some extent, on the choice of basis coordinates for G. For the same X, physically different force fields are obtained corresponding to two differently scaled sets of internal symmetry coordinates. Thus, if primes refer to quantities based on a set of internal symmetry coordinates S’scaled differently from the set S, so that S’ = bS where d is a diagonal matrix of scaling factors, then F’ = 6-‘Fd-‘, G’ = dGd, and L’ = 7J’PizX’. It is obligatory to choose X’ = r’-1’2~‘~UP’2X in order to obtain physically identical force fields by the consecutive application of Eq. (1) to the two coordinate syst,ems. For example, the choice of X = E leads, from Eq. (4)) to a symmetric GF matrix (1) and to a stationary value of Tr F; similarly, the choice of X’ = E leads to a symmetric G’F’ matrix and to a stationary value of Tr F’; but the force fields F and F’ so obtained are not physically identical. To ensure the physical identity of the two force fields (i.e., to ensure that F’ = OFd-I), it is mandatory, if X is set equal to E, to set X’ equal to rr-l'z~dUrl'z. If, on the other hand, the two coordinate systems are orthogonally related
VIBRATIONAL
FORCE
CONSTANT8
:3:3
Then it is algebraically obvious that the greatest value5 of Tr F consistent with Eq. (7) is, from Eq. (14), Tr F. = 2
I’:‘&. ,
t 161
t,he corresponding force field being Fo =
Ur-‘~8
r1 > I-2 > . . . >
i 17) r,
(1Stt)
and A* < 1~2 <
.
< A,,
( 1Sb
)
in both Eq. (16 ) and Eq. (17), so t,hat the eigenvalue products in those equations correspond to the pairing pattern obtained with both eigenvalue sequences :,I’ranged monot80nically, but in opposite order. For this particular assignment of the frequencies to the internal symmeky co-ordinates, the setting of
X=E
Cl!))
E = -r-in
( “0 1
and
defines the greatest value of Tr F obeying the newwar)- conditions, Eq. (7 1, for an extremum. If this solution, Eqs. (16-20)) also fulfills the sufficient condition for an ext’rcmum of Tr F, then Eqs. (16) and (IS) give the greatest maximal sum of t,he diagonal force co&ants corresponding to the original choice of int,crn;l] symmetry coordinates. (and therefore identically scaled), the physical equivalence of the t’wo associated forc:e fields requires that X’ be set equal to X. Thus the extremum properties of the trace of the force constant matrix and the symmetric nature of the GF matrix are independent of t,ho choice among orthogonally related systems of internal symmetry coordinates. 5 It is clear that the smallest value of TrF consistent with Eq. (7) is obtained if the frerlllency assignment shown in Eq. (18) is completely reversed, so that r1 > rs >
> r,,
and A, > A2 >
> ‘La
The subsequent argument can, mutatis mzctandis, be based upon this premiss. Actual trials are needed to indicate whether the prescription of minimal, or of maximal, values of TrF leads to previously established frequency assignments and to physically acceptable values of force constants.
FKEEMAN
34
Admittedly, the present application of the necessary condition, Eq. (7)) for extrema of Tr F could formally be made even when such extrema do not exist, but the physical interpretation would then be undermined somewhat. It is therefore of interest to inquire briefly into the nature of the sufficient condition that a solution to Eq. (7) should indeed represent an extremal value. The condition for sufhciency is quite complicated (%), being dependent upon whether a certain quadratic form, generated from a Taylor series expansion about the solution to Eq. (7)) is positive definite (existence of a minimum) or negative definite (existence of a maximum) when subject to a set of linear equations derived from the equations of constraint, Eq. (4). The resultant condition for sufficiency is expressible in terms of the alternation or continuation of sign in the expanded power series form of a related secular determinant; but the analytic application of this criterion of sufficiency is too cumbersome to be worthwhile when more than very few frequencies are involved. It has been verified that, for two frequencies, Eqs. (1618) lead to the existence of a maximal trace and that the reverse frequency assignment leads to a minimal trace,5 so that the following is always true: (“1) When more frequencies are involved, empirical computer tests for sufficiency are probably adequate. The results obtained so far, concerning force fields with stationary traces, pertain directly to any one symmetry species of a molecule and, as a corollary, to a complete molecule that has no elements of symmetry. For a molecule with some symmetry the number of distinct force constants is reduced, and the trace of the matrix j of generalized valence force constants (based on N independent internal coordinates) is Tr j = Es faa , in which some of the fuarare identical and must be included multiply in the summation. This trace is the same as that of the symmetrized force constant matrix 5 (consisting of step-matrices F along its principal diagonal) since the symmetry-factoring within 5 is accomplished by subjecting j to an orthogonal similarity transformation. Furthermore, the necessary condition for extrema of Tr j is the same as that for extrema of the traces of all the individual step-matrices F comprising 5. In other words, the conclusions previously reached for a single symmetry species may be extended to any complete molecule, provided that the multiplicity of identical force constants is correctly included in the summation of the diagonal force constants. COMPARISON
WITH
RELATED
METHODS
For the present particular solution represented by Eqs. (16-20) formation L in the relationship S = LQ
the trans(22)
between internal symmetry coordinates S and normal coordinates Q is given by L = ur”2
(231
\‘IBHATIONAL
FORCE
CONSTANTS
3.5
Hince the general formula ( 1) for L, corresponding to Eq. ( 1) , is6 L = ur”?X.
(“4)
Equation (23) k a purely kinematic defir&ion of the normal coordinate tjrans;formation in t’erms of G matrix properties. Another such definition results from the choice (%5’) of x = 0:.
i “;i 1
and corresponds to the “method of the charact,eristic set of valence coordinates”, which has been shown (7,8), by means similar to the present, to satisfy the COII,dition necessary for a stat,ionary value of c: L,, , each L,, being defined positively. This solution (assumed sufficient for a maximum) is claimed (7, 8 I to yield t,he “most diagonal” form of L. It is easily seen, because 2 L:, = Tr (Le) T.S
= Tr G = $
I’, ,
t,hat the sum of the squares of all the elements of L is independent of the choiw of X; therefore the idea is intuitively plausible t’hat t*he choice of L ZZ ur”20,
( 27 1
which maximizes c: L,, , should lead to “small” off -diagonal L elements. Ho\\.ever, perhaps the most obvious way of interpreting the phrase-“most diagonal” form of L -- is to suppose that the sum of the squares of all the off-diagonal L LK be maximaL The condition elements, CY&. L”,8, be minimal, i.e., t’hat c: for this may, by methods similar to those used in Eqs. (3-ll), he shocvli to necessitate that X be chosen so that aA = A”x,
i”S)
n,hcre A is defined as t’he matrix wibh elements A,,* , A Tb55 /JS’ IT.SY P2. 7
(“9)
The choice of X = 0, Eq. (25), requires the mat’rix UA to be symmetric, which it is generally not; hence Eq. (27) does not lead t’o the “most diagonal” form of L, at least not. in the sense defined by a minimal value for c& LFs . It is also noteworthy that the condition, Eq. (25) or Eq. (27), that gives the greatest value of Tr L arising from the necessary condit8ion for exkema of Tr L does not itself differentiate among the n! possible frequency assignmentas st,ill available for the calculation of F from Eq. l 1). III addition to the two models already mentioned for the determination of force 6Because the sign of a normal coordinate is arbitrary, there are only n! essential11 different normal coordinate transformations L = lJ@X satisfying Ey. (12) and leading to stat,ionary values of TrF. See footnot,e 2. 7 It is worth noting that Lt, = ah,/aF,.,
FREEMAN
36
constants from kinematically defined normal coordinates, a third model called the method for progressive rigidity (.c?,%-8’4) is currently in use. A summarized comparison of these three models is presented in Table I. Only the row titled “Scaling invariance” requires further comment. The effect for model I of resealing the symmetry coordinates has been discussed previously in detaiJ4 and similar considerations obtain for model II. For any harmonic force field, the physical invariance of the potential energy, kinetic energy and normal coordinates to a scaling change from symmetry coordinates S to S’ = dS (d being diagonal) demands that F’ = ddlFddl, G’ = dGd, and L’ = dL; so that, although F and G individually undergo diagonal congruence transformations (related inversely to each other), the transformation of the product GF to G’F’ = dGFd? proceeds via a diagonal similarity transformation (albeit non-orthogonal), and eigenvalues A (but not r) are left unaltered, as are the normal coordinates Q, since Q’ = L’-‘S’ = L-1d-1dS = Q. However, the symmetric property of neither GF (model I) nor L (model II) is preserved under the scaling change, in contrast to the lower triangular property of L (model III) which is preserved. Of the three models, only the method of progressive rigidity is completely independent of the scaling choice for the symmetry coordinates, and is, in this respect, theoretically preferable to models I or II. In the important special case of a second degree secular equation, U and X may TABLE
I
PROPERTIES OF FORCE FIELDS WITH KINEMATICALLY DEFINED NORMAL COORDINATES Model I
Model II L symmetric
GF symmetric L = ur”2
Transformationa Force field
F = Vr-‘Ati
L = us@7 F = Vr-“2i?1hVr-“2~
Other relations
X=E
x=8
GF = FG EL = r
GL = LG EL = G
Model III Progressive rigidity
L,, = F,, = T, s (L-‘),, (GF),,
0, r < s c:: (L-‘),,(L-‘),& = 1, . . . ,n. = 0, r < s = 0, T < s
(GO, = A, G,,= ~LmLso,r 5 s
Extremal
property
Scaling invarianceb References * The transformation
LQ. b The
TrF stationary
TrL stationary
No
No
Each F,, stationary in the (partial) system comprising only the highest T modes. Yes
(1, 4, 90,n)
(I-J, 7, 8)
(SZ-34, 40-43)
between symmetry
coordinates
S and normal coordinates
Q is S =
refers to the physical invariance or entry corresponding to “scaling invariance” otherwise of the normal coordinates, force field, and extremal property under a change in scale of the symmetry coordinates. See the text.
VIBRATIONAL
conveniently
FORCE
CONSTANTS
37
be defined by the relations
so that the second degree equivalents spect,ively , be given as e=o
of Eqs.
( 19),
(sj,
and (28)
may, re-
(maximal value of F11 + Fyi),
0 = -C#J (maximal value of Ll1 + IJ?z) (
(31) (3”)
and
tan 2e= -
(I?1rz) l”’ l,.;(rl
+ r,>
tan 24, cos 30 cos 34 > 0( minimal value of L4, + Li,).
(:3:3)
The analogous expression for the model of progressive rigidity is tan0
= -
0
112
2
tan 4.
EXAMPLES
The empirical suitability of the three models is best tested by select,ing examples, such as HzO, SOZ, and GeF, , for which the general harmonic force fields can be established accurately. (a) m’atw. The water molecule is one of the few instances in which large isotopic frequency shifts are accurately known harmonic quantities. In Table II, each row of force constants for solutions I--XIII reproduces the frequencies of the molecule(s) shown at, the extreme left of the particular row, and t’he frequent! deviations are a measure of the failure of the force field to accommodate the isotopic data exactly. Solutions XIII, XIV, and XV are the best available, and the error estimates obtained from the present work and in Refs. (do,27 j suggest that Strey’s (SC) are optimistically small. Strey prefers solution XI but it is only marginally superior to several others, and the analogous solution XII for the isotopic molecule is among the least good of the series. Only solution IV is intolerably poor. A comparison of solution I tjo II and V 60 VI shows the effect, of resealing the angular symmetry coordinate by a factor chosen (as is cust,omrrr~ but, not necessary) equal to the bond length. Because this scale factor is close to unity, and for another reasona, the effect of such resealing on the force fields is not very marked. 8 It is easy to show that, if the symmetric L matrix (solution V) is nearly diagonal (as it might well be when TrL is maximal), then a scale change will produce a nearly symmetric L’ and nearly maximal TrL’. Similarly, if in addition to being symmetric, the matrix GF is nearly diagonal, then a scale change will produce a nearly symmetric G’F’ matrix and nearly extremal TrF’. Under these conditions, the effects on the force field of a scale change will be least pronounced.
TABLE
II
COMPARISON OF KINEMATICALLY DEFINED AND ITERATIVELY REFINED FORCE FIELDS FOR WATER AND ITS ISOTOPES
Solution
Characterization of force field
fr I II III IV \ l-1 VII VIII IX X XI XII XIII XIV XV
H,O: (TrF)z,d, H20 : (TrF’):,, D,O: (TrF):,, H20: (TrF)&,, Hz0 : (TrZ)“,,, H@: (TrZ’)&,, H,O: (Z;, + L:,)% DzO : (TrZ)kaX I-I,0 : progressive rigidity8 D20: progressive rigidityg H,O: D*O: Hz0 I-I,0 1120
8.448 8.445 8.447 5.058 8.452 8.452 8.453 8.463 8.444 8.429 8.454 8.468 8.453 8.455 8.451
(F,l)!:,,, (Fll)L + 1120: exact solution’ + DrO: iteration; + DzO + HDO: iteration”
* In terms of internal coordinates 2 x P.E.
= f,.(+
Frequency deviationb (cm-‘)
Force constants” (mdyn A-l)
(TV ,Q
r?fm
f7?
0.785 0.791 0.868 4.020 0.754 0.754 0.752 0.769 0.750 0.755 0.761 0.792 0.760 0.762 0.756
-0.107 -0.110 -0.108 -3.498 -0.103 -0.104 -0.103 -0.093 -0.111 -0.126 -0.100 -0.087 -0.102 -0.099 -0.104
rLi'fczMean
Max
1.9 2.1 6.1 109.3 1.3 1.3 1.7 1.6 2.2 2.7 0.8 2.9 0.7 0.8 1.2
6 7 27 534 4 4 5 5 8 8 5 13 5 5 3
0.388 0.414 0.701 -0.108 0.159 0.157 0.140 0.299 0.044 0.841 0.236 0.441 0.225 0.246 0.168
, CX)the potential energy is given by
+ T?) + faa2 + 2frr~l~ + 2f,,(~1 + T&Y,
in which the force constant units are defined as mdyn A+ for fV and frr , mdyn for fTa , and mdyn A for fe . The quantities listed in Table II are expressed in uniform units of mdyn A-1, with ~0 = 0.95728. The unprimed kinetic energy matrix, constructed from the equilibrium parameters ~0 = 0.95728 and (~0 = 104.52” @6), is defined and scaled as in Ref. (9). The symmetry coordinates used to form P and G matrices are &@I)
= 2-“%
+ TZ),
S&41)
= cy, and
&(B1)
= 2-1’2(rl -
rz),
so that
2 x P.E.
= (f, + f,,)SP
+ 2 21’2f,S1Sz
+
fc,S,z+ (fr - f&3,‘.
b The deviation is the magnitude of the wavenumber difference between the computed harmonic frequency and the harmonic frequency derived from experiments [and quoted in Ref. (,W,96)] ; 10 such accurately known frequencies, viz., WI , wx, and WQof each of l&O, DzO and HDO and ws of T20 are used. c The matrices F and Z here refer to only the two totally symmetric vibrations of the indicated molecule. However, extrema of Tr F = f, + fr, + fa and of 2f, + .L occur together. d If this sohltion were exactly transferable from Hz0 to DZO, the skew-symmetric com0.0524 0 mutator [Glr,o , Gn,ol would vanish (1) ; in fact, [GB%o , Go,01 = 0 1 -0.0524 A primed matrix is based on resealed symmetry coordinates S’ = ds with
1
0
d= [
10 oroo.
0
00
1
1
f See Equation (33) in the text. g In the present, case of a second degree secular equation, minimal (24). h In the present case of a second degree secular equation, and GF are upper triangular matrices (24). i This solution is obtained by elementary algebra. j See Table III of Ref. (36). k See Ref. (.t?‘?‘) and Table VIII of Ref. (20). 38
this model implies that FB is this condition
implies that Z
VIBRATIONAL
FORCE TABLE
CONSTANTS
39
III
MODEL FORCE FIELDS FOR “NO* Characterization of force field
Solution
Force constants” (mdyn .-\-‘)
I__ I II III 11. 1. \‘I \I1 i-111
14N02:
(TrF)kx
14N0z: (TrF’);,, ‘%Or : (TrL) !kx 14N0t : (TrL’);:x 14N0z: (L”12+ &&xi,, 14N02: progressive rigidityd” Wo?:
(F&:x
%Or,
15NOn:
il Footnote 1)Footnote c Footnote <’ See Ref. 0 See Eq. i Footnote g Foot,notc ‘8 See I
a of Table c of Table e of Table (96). (33) in the g of Table h of Table (37).
II3 + microwave’*
12.086 12.156 12.084 12.117 12.214 11.306 12.217 / 11.033 l\*o.o5
II applies, except, that ru = l.l%iA II applies. II applies.
1.830 1.725 1.283 1.306 1.102 1.107 1.525 1.109 10.01
3.188 3.258 3.187 3.216 3.316 2.405 3.31ti 2.110 zto.05
2.634 2.483 1 633 1. CL5 1.184 0.m
2.170 0.1x1 zlA~.011
and u,, = 131”1’ (8’7 I_
test. II applies. II applies.
(b) Nitqen dioxicle. The best available force field (solut,ion VIII, Table III), incorporating microwave and infrared frequency dat,a (3’) for both of tl:r isotopic species ‘*SO2 and '%Or , is compared in Table III to the various kinematic models. The progressive rigidity model (solution VI) is clearly supcriol to t,he others which cope rather poorly with the interaction const,ants. The effects on the calculated force fields of resealing the symmetry coordinatjes are seen b! comparing solution I to II, and III to lV, to be more pronounced than for the wattrr molecule. For ‘“NO:! t’he difference bet’ween solution III, for nhich ( Lll + L,,) is maximal, and solut8ion V, for which L is “most diagonal” in the sense that (I,:, + Li1) is minimal [and (Lil + L&J maximal]. is more marked than for HZ0 (compare solutions V and VII of Table II). (c) Gew~anz’uw tet~q’ka,*ide. The general yuadrat’ic force field for t)hr !“! species is rather sharply defined by the requirement t,hat the force constants tii not, only the frequencies but also the Coriolis coefficients determined from the esperimental P-R spacings in the infrared spectrum ( 16). The value of l4 shown in Fig. 3 of the paper by J,evin and Abramo\vitz ( 16) differs from its correct valur which is given in Table I of t,hat study. The corrected values of t)he symmetr) force const,ants are those in Table IV (solution VII) of the present workY, and not those in their Table IV. Of the various model force fields in the present 9 These corrected valnes from Dr. I. W. Leviu.
were supplied
in a private
comrnllnicatiou
(Jalluary
191%)
40
FREEMAN TABLE
IV
MODEL FORCE FIELDS FOR GEF~ (Fz SPECIES) Solution
Characterization of force field
I II III IV V VI
(TrL’)mex (TrL):,, Progressive (F’a&ex
VII
Exp. Coriolis
Force constants” (mdyn A-l) FIR
(TrF’)max
6.102
(TrF)“,ax
3.987
rigidity
6.176 6.105 5.671 6.266 5.87 f0.05
& -1.813 -2.455 -0.848 -0.699 -0.160 -1.288 -0.32 f0.05
F’46 0.787 1.892 0.362 0.328 0.271 0.507 0.27 10.01
Coriolis coefhcientsb
r3a -0.38 -0.42 -0.04 +o.oa f0.26 -0.24 +O.ZOd
.c44 i-O.88 +0.92 f0.54 $0.48 +0.24 +0.74 i-0.354
&The force constant and kinetic energy matrices are based on symmetry coordinates Sa’ = ,14(ri - T%- r1 + ~4) and Sa’ = r02-‘@ (01~~ - o[i4) defined in terms of internal coordinates ri and ~lij and,th,ez equilibrium paramet:rs (16) TO = 1.67A, 010 = 109”28’. The potential energy is M(Fd% + 2F&‘S4’ + F&I 1 in which the force constant units are mdyn A-‘. b These are calculated from the following formulae, which are essentially those of Ref. (58), but with some signs changed ty confyrm to t$e usage of Ref. (16) : (33+ ska= 1 and r33 = (A, - A&i [Al + 5s 114- pp(Fz3 f Fad - 2F34)1, in which /*p is the reciprocal of the atomic mass of fluorine. c An unprimed matrix is based on the resealed symmetry coordinates SO = S3’ and sq = r,’ Sa’. d The experimental Coriolis coefficients obey the sum rule only approximately. See Ref. (16) and the text.
Table IV, the model of progressive rigidit’y gives the most satisfactory results, especially as regards the interaction constant and the Coriolis coefficients”. The marked differences between solutions I and II, and, to a lesser extent, between solutions III and IV show the dependence of these kinematic models on the scaling of the symmetry coordinates. (d) Othey molecules. The model with symmetric GF and extremal Tr F (Table I, model I) might be expected, from Tables III and V of Ref. (SC), to yield fairly satisfactory force constants for H&S (and D2S) and HzSe (and D2Se) but not for SO, or C102 . Plausible results are obtained for C&H, (30) and, it is stated (31)) for the Al, species of CeH6 ; but in neither case were deuterium substituted molecules studied. Model I is claimed (31) to result in unconventional force fields for RF,, BCl, , and SF,. lo For SiF4 , the force fields of Duncan and Mills (14) and of Shimanouchi et al. (20) are based on an estimated value of 53 = 0.63 for which no great reliability was claimed (14). The more recent determination by Levin and Abramowitz (16) gives 5‘3 = 0.49, which value corresponds to a force field in good agreement with the model of progressive rigidity. However, the observed value of rh = -0.12, which leads to a sum rule inconsistency, is still an unresolved difficulty.
VIBRATIONAL
FORCE
CONSTANTS
-41
The model with symmetric L and extremal Tr L (Table I, Model II) has been t#ested for the alkyl (3) and silyl (3, 4) halides; the results are reasonable compared to hybrid orbital force fields (12, 39)) although certain insen&ive intcraction constank constrained t’o remain zero in t’he hybrid orbital force firld att,ain values up to 0.S mdynA_’ in model II. The agreement bet#n-een the :I(‘curately known force constants (13) and model II (7, 8) is good for SiH, and fair for SiD, . Model II yields force constant,s that are reasonable for H,S C:uld D?S) and HSe (and D?Se) but poor for SO2 and Cl02 [Table V of Ref. ( .Sj ) I. Evaluation of the application to C?H&l (8) and HaBOa i A4) is dificult If it is remembered that the model of progressive rigidit,y and t)he :~tt:tinment of a minimal value for Fzf are equivalent’ for a second degree secular eq~l:~tion, :I variety of molecules becomes available for which this model may be evaluated crit,ically. The model is a good approximat’ion for the following: SO? (.I/; i ; H,S (.%);H,Se (36); CHJ, CDJ, CT4 (13);XH4, XD,(X = Si, Ge) (13);l~,‘spc~+~s of XHs , XD,(X = S, I’, As, Sb) (13); E species of AsI::{ ( 16) ; F1,, species of XFs(X = Se, Te, W) (17, 28‘); GeCl, (,/to, .(l 1. The model is a fair approsim:ltion for the following: ClO, (36); E species of BX,(X = F, Cl, Br, I) (1~5); E sp(xcies of PI?, (16‘); CX,(X = P‘, Cl, Br) (I/,, II?, ,$O, ,il ). The model is a poor :Lpprosimation for the following: A1 speicies of XH, , XD,( X = S, I’, t\s, Sb) ( I,Z) ; I:’ species of NF, (16) ; Fl,, species of SIC0 (17). In all of the above C:LSESI hc force fields are rather narrowly const)rained by experimentally known (‘oriolis coefficients, centrifugal distortion coefficients, or mean square vibraLtional amplitudes. The model of progressive rigidity has also been applied to the follo\ving : I-X4(X = F, Cl, Br; I- = I’b, Zr, Hf) (42, @) ; A1 species of CH,C’l, (.I,” ) ; A-1 ‘, species of C2H4 (3%‘); CeH,, (33). In t’hese cases, ho\vever, t)hc true force ti~ltls :ire not so accurately known. (c) Conclusion. From the limited number of examples surveyed, the provisional conclusion is drawn that the model of progressive rigidit,?- is the most satisfactory of the t’hree kinematic models examined in this paper. It is noted. however, that most of the molecules t)o which crit’icxl t,ests can be applied possc~s symmetry species containing only a fen- well-spaced frequencies, and that this is likely to be a necessary, though not’ a sufficient 1condition for success of the moclr~l of progressive rigidit,v. SUMMA11 Y ( a) The criterion that the sum of the diagonal force constants of a gencrnlized valence force field fulfill the condition necessary for att#ainment, of a stutiollary value, consistent with elimination of the arbitrariness inherent in the usual unrestricted formulation of t’he inverse secular problem of determining ~.$L(?L + 1) svmmetry force constants from t’he equilibrium geometry andonlJ !L vibrational frequencies, leads to ‘n! solutions for the force field, E(l. (13). Thr force fields so determined are independent of the choice of orthogonall\related
FREEMAN
42
coordinate systems but are dependent on the scaling chosen for the internal symmetry coordinates. These n! solutions, previously discussed in another context ( 30,31) are further characterized (1) by the requirement that their GF matrices be symmetric. The circumstances under which these solutions are transferable to an isot,opic molecule have also been examined (I). (b) A narrowing of the scope of the above criterion to the condition necessary for the greatest extremum of the trace of the F matrix leads to a unique solution for the force field, Eq. (17). The appropriate normal coordinate transformation, Eq. (23), is specified in terms of the eigenvalues and eigenvectors of the G matrix. The corresponding assignment of the vibrational frequencies to a sequence of the chosen internal symmetry coordinates is determined by the sequence of eigenvalues of the G matrix, in accordance with Eq. (18). The reverse of the vibrational assignment represented by Eq. (18) corresponds to the condition necessary for the smallest extremum of the trace of the F matrix. (c) The above conclusions concerning force fields with stationary traces relate specifically to a single symmetry species of a molecule. Their extension to any complete molecule is indicated and involves 110 novel features. (d) The analytic basis and physical interpretation of one method of determining molecular vibrational force constants from kinematically defined normal coordinates has been examined in detail and compared to two other such methods, viz., the method of the characteristic set of valence coordinates, and the method of progressive rigidity. The last method has two advantages: it alone is completely independent of the scaling of the internal symmetry coordinates, and, it appears, from an empirical examination of a limited number of examples, to be the most satisfactory of the t’hree kinematic models tested. Any of the three models yields real and readily calculable sets of force constant’s which may be of use in cases where Coriolis coefficients etc., are not available. RECEIVED:
October 25, 1967 REFERENCES
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