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Molecular translation—rovibronic hamiltonian. II. Linear molecules
ChemicalPhysics75 (1983) North-Holland
Publishing
157-164 Company
MOLECULAR TRANSLATIONAL-ROVIBRONIC
HAMILTONIAN.
II. LINEAR MOLECULES *
Reza ISLAMPOUR Dcpar!ment Tallahassee, Received
and Michael KASHA
of Chemistry and Institute FIorida 32306. USA
18 October
of Molecular
Biophysics,
Florida State
Universit_v.
1952
The transformation of the translntionnl-rovibronic (non-relativistic) hamiltonian from cartesian to generalized nates by the quantum-mechanical path outlined in the previous work has been carried out for linear molecules.
1. Introduction
of generalized [2.3]:
The present work is a continuation of the previous investigation [l] (hereafter referred to as I) in which an attempt is made to set up the hamiltonian for a non-linear polyatomic molecule in terms of molecular (translational, rotational, vibrational and electronic) coordinates. Because of some peculiarities, and in order to avoid confusion, we excluded the linear molecules from that work. However, the present extension is developed in strict analogy to the previous case, with the necessary variations for the linear geometry. In I we used the following two fundamental theorems: (i) The hamiltonian operator for a coliection of Coulomb-interacting particles in terms of a set of labfiied (LF) coordinates ** R = (REX) is given by: AR = -4Ii’
F
V’lrnh + V(R),
coordinates
coordi-
4 = {q,}, eq. (1) becomes
where
(3) g = (det lgT”i)-l and pq is a weighting factor in the volume elements. (ii) The components of the total angular momentum operator for the collection of particles with respect to the origin of the LF frame are defined by [31
(1)
where P(R) is the potential energy of the entire system of particles_ When expressed in terms of a real set
when expressed in terms of the generalized nates 4. then (4) becomes
* This work was sponsored by a contract between the sion of Biomedical and Environmental Research of Department of Energy and Florida State University, tract No. DE-AS-5-7SEV05855. ** In this work the index h refers to a general particle, the indicesf,g, II, refer to a general direction of the frame ases.
where [&f: LJ~]is the commutator between JI?~ and qT_ Note that the forms (5) still apply to the wavefunctions in R-space_ The set {s7] for a linear molecule composed of 1%’nuclei and 11electrons includes four subsets of {3 translational), 11 rotational), {3V-5
0301-0104/83/0000-0000/S
Divithe US. Conand LF
03.00 0 1983 North-Holland
coordi-
R Islampour, Al. Kasha / ~andationai-rovibronic
158
vibrational} and {311 electronic) variables. The purpose of this paper, analogous to I, is therefore to derive here the hamiltonian of a linear molecule in terms of the molecular coordinates by using the above theorems.
2. Transformation
equations
to generalized
coordinates
In order to separate the translational, rotational, vibrational and electronic variables as completely as possible, we may transform the LF Cartesian coordinates of the nuclei and electrons in the linear molecule to a moving body-fiied (BF) frame through analogous steps presented in 1. If the position of a particle (an electron or a nucleus) is R,x,in the LF frame and r~ in the BF frame, then the general transformation equation may be formulated as: rA=Rh-R-d.
(6)
where R is the position of the molecular mass with respect to the LF frame R = F
center of
mkRJM,
(7)
Pj = -r:j
hamiltomlm. II
+ Rj -
C JJ.?iRi/JJlN i
for the nuclei. Where the displacements pj = rj - rtj describe the deviations of the nuclei from the equilibrium positions rzqi_Of course
C??'ipj'O . (12) i We assume the equilibrium position of each nucleus in the case of the linear molecule to be on the z axis of the BF frame. To specify the orientation of the z axis (molecular axis) in space, we require two rotational variables: these two variables (and only two) are defined implicitly by the Sayvetz-Eckart [4,5] conditions as cJJZj~$~Xj=O
_
(13)
j
We see that the Sayvetz-Eckart conditions are necessary and sufficient for our purpose but not sufficient to define the BF frame completely. Substituting for pj from (1 l), with the help of the equilibrium form of (8), then (13) gives C,njrtj i
XRi=O.
(14)
with M = ZXJJZ~, the total mass of the molecule. Here d is a vector from the molecular center of mass to the
The 3N-5 vibrational nates) are defined by
nuclear center of mass. Since the origin of the BF frame is at the nuclear center of mass *
pj=??l‘lClx- ].kok
~~Ilj~j=O,
F
l?liRi/??ZN
(1%
(9)
C ntj-lb,k
- o,r = 6~,
i
Flj,k=O
9
(16)
(17)
(10) Fr:j
for the electrons
coordi-
where the coefficients 4,X_which have constant components with respect to the BF frame satisfy the orthogonality relations [6]
where JnN is the total mass of the nuclei. Introducing (9) into (6) we may write rs = R, -
9
variables Qk (normal
GO
therefore, d = -R + C ?niRi/mN , i
(11)
XIj,k=O
-
(1%
and
* In this work (as in I) the indices i, j refer to the nuclei; s, r to the elections; k, 1, m, n to the vibrational variables,and u or u is a general rotational variable.
The inverse of (15) is Qx_= Clj,ki
pj _
11%
159
R. Islampour, IV. Kasha / 7Fanslational-rovibronic hamiltonian II
Finally, the transformation equations ular coordinates may be summarized as
to the molec(27)
R = c q,RA/M, A
(20)
C,IvrtjXRj=O, i
(21)
or dc,idu
= g
,
&&fl(e&~
(28)
where X,, are some coefficients. into (26) we obtain easily
Introducing
(28)
(22d ZhlaRfi
Pj= -r$ + Rj
-
C i
ntiRi/tnN
,
(23)
i
3.
(22b)
The calculation
of g’” and g
The calculation of g70 requires the evaluation of the derivative of the molecular coordinates with respect to the LF frame coordinates and upon insertion into (3). From (20) aR&Rfx
= Fn~iW1 Sfg .
(24)
The evaluation of au/aRfi requires the same mathematical trick used for non-linear molecules, as follows_ Since the unit vectors * E& along the BF frame axes are orthogonal, we can write
a&,jaRfi=E,xaj-.
(25)
where 3; are some vectors to be determined. If eg are the unit vectors along the LF frame axes, then (25) can be written as eg - LkJaRfi
= eg
&
c&(X-~)~~
,
(29)
where c$= is the component of o>along the Q axis and X-’ is - reciprocal to X. (The foml of matrices (e&, is given in the appendix of I.) The matrix C contains three independent rotational variables of which two are sufficient to specify the orientation of a linear configuration in space. Therefore. we may let the third variable enter into the calculation as an arbitrary parameter; let this arbitrary parameter be called X, then from (29) ax/aRfi = 0 = c
Q
c&(X-~)~
_
WQ
Thus
where A, = (X-‘),/(X-‘)l,
(32)
-
From (29) and (3 1) we may write azz/aRfi = c w&(Y-~)~~ Q
(a =x.
y) ,
(26)
where C,, = eg - E? is an element of the direction cosine matrix C. Since C is orthogonal, the CT(aC/an) is a skew symmetric matrix and can be written in terms of skew symmetric matrices eOpy = (e,)h as follows
(Y-&
= (x-t),
-A&X-$,
.
cated.
overx,y,
z unless
otherwise esplicitly
(34)
(The form of matrix Y -’ is given in the appendix, along with X-l for purposes of comparison_) The derivative of the vibrational variables Qx_with respect to the nuclear coordinates Rfi can be written as
aQdaRfi= C(aQ,lap,j)(ap,laRfi) ; * *, p, ‘y, ___go
(33)
for the two rotational variables I[. where the 3 X 3 matris Y -’ is defined by
-(ETX wj)
= F (ac,,/all) WaRfi,
= c
ai
indi-
from (22a)
(35)
R Idampour. hf Kasba / Translational-rovibronic
160
aQdapaj=
faj.k
hamiltonian. II
(36)
(43)
and from (22b) (W
+ mi * [(Pj + r29_>X =a] ,
(37)
where (25) has been used. Introducing (36) and (37) into (35) and making use of (17) and (18) we get
r”=IO++Q~_
(45)
Introducing aQklaRfi
= Ui.k - ej-) + 03 - pcm/cQrn
where the coriolis defined by [5] &A_’
C,Z,zrl(Ij,,n i
coupling
coefficients
,
(W
&,.& are
XIj,k) -
(3%
Finally, from (23) the derivative of the electronic variables with respect to the LF frame coordinates are i+ys/aRfr = Gt Cf,
ti;e=
W:z6,
Therefore, expressions deduce the ponents of cc&=0 i
(43)-(45)
into (42), we get
- 1?Zi(Z”)-‘Eu * (rZqiX ej-) _
(46)
(31) and (46) complete the required for the components of o>_ It is easy to fo!lowing identities for the x andB com0;: (a=x,_V)
9
(47)
Wa)
(48)
and ar,.JaRfi = -tjlitrl&‘Cfy + 0;
- (rs X E-& ,
Wb)
where (25) has been used. To determine the vector oj completely, we need to find expressions for the two other components because by (3 1) the third component is expressed in terms of the two other components. Taking the derivative of (21) with respect to Rfi, we obtain
j Inserting (15) into (11) and substituting after multiplying by (- Ed), we get
(41) for Rj in (41),
where the vector identity (A X B) X C= (,4 - C) B (B - C) A, (18) and the equilibrium form of (8) have been used. This expression permits the definition of the following quantities:
(a,P=x,u)
-
(4%
Now, by careful insertion of the derivatives (24), (33), (38) and (40) into (3) and making use of (17) and (18) and (47)-(49), all elements g’” can be calculated. The results are collected in table 1. In this table, only (Y= x, y. Taking the table as a determinant, we can simpIify it by column and row operations in the following order (in these operations only y =x, v): (i) multiplication of column u by Z, Y,<$Q, and subtracting from column I; (ii) multiplication of column u by &,n.~rYW&Q,l and adding to column I; (iii) multiplication of column u by ZWe,prct Y, and subtracting from column (St); (iv) multiplication of coIumn u by CWez~~rgtA,YW and adding to column (6r); (v) multiplication of row u by Zrunt YuJ&kQm and subtracting from row k; (vi) multiplication of row u by Z,,A7Y&&Qm and adding to row k; (vii) multiplication of row u by IZW,.,e,r,,sYU, and subtracting from row (7s);
R Isla~npo~r, hf Kashn / ‘12nnslatiolml-rol~ibrollic hamiltonian Table 1 Matrix of the contmvariant
II
161
metric tensor elementsgTu
T
R
V
Q gh = M-’ sgh
gg”=o
Q qf =o
E
I
1
The row-column
labels
T. R. V. E.
deslgnale
lronslniionol.
rototvmol.
0 ---
] /I[(Y-l)T(Y-l)] 1 0 +____~__~~_~~~~~~~~
0 ---
! 0 ,________1___~_____ :
0 :
i 0
---
0
:
g = 41f3,p1311(yyp-2 where (Y’)-’ appendix)
= det I[Y-l),,
(50)
I with cr = _r,~*. (See
in terms of the
: 0
’
0
:
0
ln-1
:
, 13,~
To express the generalized angular momentum (5) in terms of molecular coordinates, WC need to calculate the commutators between JOT and those coordinates as follows. Using (4).
J
]j$-,
+I
= ifi F qg~dG,x7
where, e.g., 13 is a 3 X 3 unit matrix and nz is the mass of the electron. Therefore,
or in vector form
det ]gTu I = Mm3mm3”(
[hlf,Rh]
Y’)-*p’
.
4. Total angular momentum molecular coordinates
0
13,*-5 ’
~----_---7---7_-___
electronic
or
(viii) multiplication of row u by Z,, errs r+, Yur and adding to row (7s); (ix) writing the explicit form of nz-16y6 6rr + nts16,6 by which the determinant can be further simplified by column operations. The matrix of the simplified determinant is M-113 i 0 --+__-__---7-_-~--___
vibrotionol.
,
=ih(RhXef)_
(51)
(52)
R Islampour. hf. Kasha / lFandational-rovibronic
162
The commutator implies that
between “irand
(21) by using (52)
PQJ,EzI= Wk Xer)
(53)
hamilfonian. II
total angular momentum of the molecule about the center of mass. We shall continue to use the same notation Nf for the angular momentum operators about the center of mass:
Or
n;lr= g c ci
efgll
Rgi
da -
IfA?fis nents&
Therefore, [&f, 8 p] = itlep X ef - ifi C Cr,A&p Q
X E=) ,
[fif,
= 3 T efghRh ,
241 = ifi I$
Introducing
Cf&Y_‘),
(56)-(59)
(Y-l),
&+A,(&+&r)].
(63)
projected along the BF frame axes, the compoare then (64)
or
(56) ,
q
(55)
where (25) has been used. Using the transformation equations (20), (22), and (23) with the help of the above results, it is now easy to verify that ii’+-, &I
Cf+
(54)
(@=x2).
= -C’&
(57)
+ @I
cP erapAp,
(65)
where (55) has been used. Here (64) comes from the fact that fif does not commute with Cfa as seen from (55). We see that the only source of angular momentum along the z axis of the BF frame is the z components of the electronic and so-called vibrational angular momentum (as it ought to be). If p4 is chosen to be the determinant Y’ and [&l_c] = 0, then th e f orms (64) and (65) persist for the q-space.
into (5) gives 5. Molecular
(60) where the electronic angular momentum e, and socalled vibrational angular momentum c(I are defined by [6]
(61)
translational-rovibronic
hamiltonian
Now we substitute for g from (50) and for the elementsg’O from table 1 in the generalized hamiltonian (2) In the first place, as in the non-linear case, this again appears to be a complicated task (which it likewise is!). However, we may have similarly a remarkable simplification if: (i) pq in (2) is chosen to be the determinant * Y’; in turn (2), after expansion, takes the analogous form to that for the non-linear case:
and
(62) The first term in (60) is the angular momentum of the center of mass of the molecule with respect to the origin of the LF frame, and the second term is the
* pq may be taken minus the determinant Y’ in the volume elements_ However, this does not produce any change in the present calculation.
R Islampour.
M. Kasha / 7hnslational-rovibronic
hamiltonian
163
II
6. Conclusion
where (50) has been used. (ii) (65) is expressed in terms of the angular momentum operators defined by (61), (62) and (65). (iii) [cm, ~1 = 0, which it actually is [6]. Following the above remarks, and recognizing that p is only a function of the Q and both Y’ and A, are only functions of rotational variables, the molecular translational-rovibronic hamiltonian for the linear molecule is derived to be
Appendix:
A= -tw/2M +
+
( (-fn’
-12
In this paper our previous derivation of the generalized coordinate complete (non-relativistic) molecular hamiltonian for non-linear molecules is extended to the linear molecule case. The derivation is parallel to the previous one, with the initial difference of course being the simplified molecular coordinate disposition along the z auk. In previous work, the derivation of the complete rotation-vibration-electronic hamiltonian has been given for the linear case of diatomic molecules [7- 111. The generai conclusions presented in our previous paper 111 apply equally well to both non-linear and linear molecules, since the two derivations are so analogous.
c v?/21,1- A2c St
S
F
a2//aQi
v,
-
VI /21?1 N
)
)
The matrix
Y-’
For the eulerian angles 0,@, x the matrix X-t defined in the appendi__ of 1 IO be
[sin x x-l=-
I
-csc 0 cos x
cosx
csc 0 sin x
Lo
is
cot 0 cosq -cot
0
0 sin x f. I
(A-1)
J
Then (34) gives Y-‘=-fcosxKsin x
-csc cscOsin)i e cos x 0
1 0 where (Y,0 =x,J’; r= {r,,} denotes the set of electronic variables and Q = {Qk} the set of vibrational variables. The term I/ is much simpler than for the non-linear case [l], and it is absorbed in the fourth term of (67). In (67) the first three terms represent the translational, electronic and vibrational contributions to the total kinetic energy of the molecule, respectively_ The fourth term is the contribution of rotational kinetic energy and the energy of interactions (between vibrations and rotations plus electronic motions) to the total kinetic energy of the molecule_ The coordinatedependent part of the kinetic energy U is also absorbed in the fourth term. Finally, the last term is the potential energy of the entire system of electrons and nuclei.
Therefore,
(y’)-‘=
the determinant
-
0 1 01 1.
(A-2)
Y’ is
Isinx
(A-3)
cos x
or Y’ = -sin
8.
References [ 11 R. Islampour and 51.Esshs. Chem. Pi]\-s. 74 (1983) 67. [1] 9. Pod&l+-,
Phys. Rev_ 32 (1928)
8 12.
164
R. Islampour. M. Kasha / Translational-rovibronic
131 EC. Kemble,The fundamental principles of quantum mechanics (Dover. New York, 1937. 1958) p. 227. [4] A. Sayvetz. J. Chem. Phys. 7 (1939) 383. 15 I C. Eckart, Phys. Rev. 47 (1935) 552. [6] J.K.C. Watson. Mol. Phys. 19 (1970) 465. 171 R. de L. Kronig. Band spectra and molecular structure (Cambridge Univ. Press, London, 1930) pp. 6-16.
itami&onian 11
[S] J.H. van Vleck, Phys. Rev. 33 (1929) 467. [Y] G.H. Diekc, Phys. Rev. 47 (1935) 870. [IO] P.R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, 1979) p. 129. [l l] R. T Pack and J.O. Hirschfelder, J. Chcm. Phys. 49 (1968) 4009.
Publishing
157-164 Company
MOLECULAR TRANSLATIONAL-ROVIBRONIC
HAMILTONIAN.
II. LINEAR MOLECULES *
Reza ISLAMPOUR Dcpar!ment Tallahassee, Received
and Michael KASHA
of Chemistry and Institute FIorida 32306. USA
18 October
of Molecular
Biophysics,
Florida State
Universit_v.
1952
The transformation of the translntionnl-rovibronic (non-relativistic) hamiltonian from cartesian to generalized nates by the quantum-mechanical path outlined in the previous work has been carried out for linear molecules.
1. Introduction
of generalized [2.3]:
The present work is a continuation of the previous investigation [l] (hereafter referred to as I) in which an attempt is made to set up the hamiltonian for a non-linear polyatomic molecule in terms of molecular (translational, rotational, vibrational and electronic) coordinates. Because of some peculiarities, and in order to avoid confusion, we excluded the linear molecules from that work. However, the present extension is developed in strict analogy to the previous case, with the necessary variations for the linear geometry. In I we used the following two fundamental theorems: (i) The hamiltonian operator for a coliection of Coulomb-interacting particles in terms of a set of labfiied (LF) coordinates ** R = (REX) is given by: AR = -4Ii’
F
V’lrnh + V(R),
coordinates
coordi-
4 = {q,}, eq. (1) becomes
where
(3) g = (det lgT”i)-l and pq is a weighting factor in the volume elements. (ii) The components of the total angular momentum operator for the collection of particles with respect to the origin of the LF frame are defined by [31
(1)
where P(R) is the potential energy of the entire system of particles_ When expressed in terms of a real set
when expressed in terms of the generalized nates 4. then (4) becomes
* This work was sponsored by a contract between the sion of Biomedical and Environmental Research of Department of Energy and Florida State University, tract No. DE-AS-5-7SEV05855. ** In this work the index h refers to a general particle, the indicesf,g, II, refer to a general direction of the frame ases.
where [&f: LJ~]is the commutator between JI?~ and qT_ Note that the forms (5) still apply to the wavefunctions in R-space_ The set {s7] for a linear molecule composed of 1%’nuclei and 11electrons includes four subsets of {3 translational), 11 rotational), {3V-5
0301-0104/83/0000-0000/S
Divithe US. Conand LF
03.00 0 1983 North-Holland
coordi-
R Islampour, Al. Kasha / ~andationai-rovibronic
158
vibrational} and {311 electronic) variables. The purpose of this paper, analogous to I, is therefore to derive here the hamiltonian of a linear molecule in terms of the molecular coordinates by using the above theorems.
2. Transformation
equations
to generalized
coordinates
In order to separate the translational, rotational, vibrational and electronic variables as completely as possible, we may transform the LF Cartesian coordinates of the nuclei and electrons in the linear molecule to a moving body-fiied (BF) frame through analogous steps presented in 1. If the position of a particle (an electron or a nucleus) is R,x,in the LF frame and r~ in the BF frame, then the general transformation equation may be formulated as: rA=Rh-R-d.
(6)
where R is the position of the molecular mass with respect to the LF frame R = F
center of
mkRJM,
(7)
Pj = -r:j
hamiltomlm. II
+ Rj -
C JJ.?iRi/JJlN i
for the nuclei. Where the displacements pj = rj - rtj describe the deviations of the nuclei from the equilibrium positions rzqi_Of course
C??'ipj'O . (12) i We assume the equilibrium position of each nucleus in the case of the linear molecule to be on the z axis of the BF frame. To specify the orientation of the z axis (molecular axis) in space, we require two rotational variables: these two variables (and only two) are defined implicitly by the Sayvetz-Eckart [4,5] conditions as cJJZj~$~Xj=O
_
(13)
j
We see that the Sayvetz-Eckart conditions are necessary and sufficient for our purpose but not sufficient to define the BF frame completely. Substituting for pj from (1 l), with the help of the equilibrium form of (8), then (13) gives C,njrtj i
XRi=O.
(14)
with M = ZXJJZ~, the total mass of the molecule. Here d is a vector from the molecular center of mass to the
The 3N-5 vibrational nates) are defined by
nuclear center of mass. Since the origin of the BF frame is at the nuclear center of mass *
pj=??l‘lClx- ].kok
~~Ilj~j=O,
F
l?liRi/??ZN
(1%
(9)
C ntj-lb,k
- o,r = 6~,
i
Flj,k=O
9
(16)
(17)
(10) Fr:j
for the electrons
coordi-
where the coefficients 4,X_which have constant components with respect to the BF frame satisfy the orthogonality relations [6]
where JnN is the total mass of the nuclei. Introducing (9) into (6) we may write rs = R, -
9
variables Qk (normal
GO
therefore, d = -R + C ?niRi/mN , i
(11)
XIj,k=O
-
(1%
and
* In this work (as in I) the indices i, j refer to the nuclei; s, r to the elections; k, 1, m, n to the vibrational variables,and u or u is a general rotational variable.
The inverse of (15) is Qx_= Clj,ki
pj _
11%
159
R. Islampour, IV. Kasha / 7Fanslational-rovibronic hamiltonian II
Finally, the transformation equations ular coordinates may be summarized as
to the molec(27)
R = c q,RA/M, A
(20)
C,IvrtjXRj=O, i
(21)
or dc,idu
= g
,
&&fl(e&~
(28)
where X,, are some coefficients. into (26) we obtain easily
Introducing
(28)
(22d ZhlaRfi
Pj= -r$ + Rj
-
C i
ntiRi/tnN
,
(23)
i
3.
(22b)
The calculation
of g’” and g
The calculation of g70 requires the evaluation of the derivative of the molecular coordinates with respect to the LF frame coordinates and upon insertion into (3). From (20) aR&Rfx
= Fn~iW1 Sfg .
(24)
The evaluation of au/aRfi requires the same mathematical trick used for non-linear molecules, as follows_ Since the unit vectors * E& along the BF frame axes are orthogonal, we can write
a&,jaRfi=E,xaj-.
(25)
where 3; are some vectors to be determined. If eg are the unit vectors along the LF frame axes, then (25) can be written as eg - LkJaRfi
= eg
&
c&(X-~)~~
,
(29)
where c$= is the component of o>along the Q axis and X-’ is - reciprocal to X. (The foml of matrices (e&, is given in the appendix of I.) The matrix C contains three independent rotational variables of which two are sufficient to specify the orientation of a linear configuration in space. Therefore. we may let the third variable enter into the calculation as an arbitrary parameter; let this arbitrary parameter be called X, then from (29) ax/aRfi = 0 = c
Q
c&(X-~)~
_
WQ
Thus
where A, = (X-‘),/(X-‘)l,
(32)
-
From (29) and (3 1) we may write azz/aRfi = c w&(Y-~)~~ Q
(a =x.
y) ,
(26)
where C,, = eg - E? is an element of the direction cosine matrix C. Since C is orthogonal, the CT(aC/an) is a skew symmetric matrix and can be written in terms of skew symmetric matrices eOpy = (e,)h as follows
(Y-&
= (x-t),
-A&X-$,
.
cated.
overx,y,
z unless
otherwise esplicitly
(34)
(The form of matrix Y -’ is given in the appendix, along with X-l for purposes of comparison_) The derivative of the vibrational variables Qx_with respect to the nuclear coordinates Rfi can be written as
aQdaRfi= C(aQ,lap,j)(ap,laRfi) ; * *, p, ‘y, ___go
(33)
for the two rotational variables I[. where the 3 X 3 matris Y -’ is defined by
-(ETX wj)
= F (ac,,/all) WaRfi,
= c
ai
indi-
from (22a)
(35)
R Idampour. hf Kasba / Translational-rovibronic
160
aQdapaj=
faj.k
hamiltonian. II
(36)
(43)
and from (22b) (W
+ mi * [(Pj + r29_>X =a] ,
(37)
where (25) has been used. Introducing (36) and (37) into (35) and making use of (17) and (18) we get
r”=IO++Q~_
(45)
Introducing aQklaRfi
= Ui.k - ej-) + 03 - pcm/cQrn
where the coriolis defined by [5] &A_’
C,Z,zrl(Ij,,n i
coupling
coefficients
,
(W
&,.& are
XIj,k) -
(3%
Finally, from (23) the derivative of the electronic variables with respect to the LF frame coordinates are i+ys/aRfr = Gt Cf,
ti;e=
W:z6,
Therefore, expressions deduce the ponents of cc&=0 i
(43)-(45)
into (42), we get
- 1?Zi(Z”)-‘Eu * (rZqiX ej-) _
(46)
(31) and (46) complete the required for the components of o>_ It is easy to fo!lowing identities for the x andB com0;: (a=x,_V)
9
(47)
Wa)
(48)
and ar,.JaRfi = -tjlitrl&‘Cfy + 0;
- (rs X E-& ,
Wb)
where (25) has been used. To determine the vector oj completely, we need to find expressions for the two other components because by (3 1) the third component is expressed in terms of the two other components. Taking the derivative of (21) with respect to Rfi, we obtain
j Inserting (15) into (11) and substituting after multiplying by (- Ed), we get
(41) for Rj in (41),
where the vector identity (A X B) X C= (,4 - C) B (B - C) A, (18) and the equilibrium form of (8) have been used. This expression permits the definition of the following quantities:
(a,P=x,u)
-
(4%
Now, by careful insertion of the derivatives (24), (33), (38) and (40) into (3) and making use of (17) and (18) and (47)-(49), all elements g’” can be calculated. The results are collected in table 1. In this table, only (Y= x, y. Taking the table as a determinant, we can simpIify it by column and row operations in the following order (in these operations only y =x, v): (i) multiplication of column u by Z, Y,<$Q, and subtracting from column I; (ii) multiplication of column u by &,n.~rYW&Q,l and adding to column I; (iii) multiplication of column u by ZWe,prct Y, and subtracting from column (St); (iv) multiplication of coIumn u by CWez~~rgtA,YW and adding to column (6r); (v) multiplication of row u by Zrunt YuJ&kQm and subtracting from row k; (vi) multiplication of row u by Z,,A7Y&&Qm and adding to row k; (vii) multiplication of row u by IZW,.,e,r,,sYU, and subtracting from row (7s);
R Isla~npo~r, hf Kashn / ‘12nnslatiolml-rol~ibrollic hamiltonian Table 1 Matrix of the contmvariant
II
161
metric tensor elementsgTu
T
R
V
Q gh = M-’ sgh
gg”=o
Q qf =o
E
I
1
The row-column
labels
T. R. V. E.
deslgnale
lronslniionol.
rototvmol.
0 ---
] /I[(Y-l)T(Y-l)] 1 0 +____~__~~_~~~~~~~~
0 ---
! 0 ,________1___~_____ :
0 :
i 0
---
0
:
g = 41f3,p1311(yyp-2 where (Y’)-’ appendix)
= det I[Y-l),,
(50)
I with cr = _r,~*. (See
in terms of the
: 0
’
0
:
0
ln-1
:
, 13,~
To express the generalized angular momentum (5) in terms of molecular coordinates, WC need to calculate the commutators between JOT and those coordinates as follows. Using (4).
J
]j$-,
+I
= ifi F qg~dG,x7
where, e.g., 13 is a 3 X 3 unit matrix and nz is the mass of the electron. Therefore,
or in vector form
det ]gTu I = Mm3mm3”(
[hlf,Rh]
Y’)-*p’
.
4. Total angular momentum molecular coordinates
0
13,*-5 ’
~----_---7---7_-___
electronic
or
(viii) multiplication of row u by Z,, errs r+, Yur and adding to row (7s); (ix) writing the explicit form of nz-16y6 6rr + nts16,6 by which the determinant can be further simplified by column operations. The matrix of the simplified determinant is M-113 i 0 --+__-__---7-_-~--___
vibrotionol.
,
=ih(RhXef)_
(51)
(52)
R Islampour. hf. Kasha / lFandational-rovibronic
162
The commutator implies that
between “irand
(21) by using (52)
PQJ,EzI= Wk Xer)
(53)
hamilfonian. II
total angular momentum of the molecule about the center of mass. We shall continue to use the same notation Nf for the angular momentum operators about the center of mass:
Or
n;lr= g c ci
efgll
Rgi
da -
IfA?fis nents&
Therefore, [&f, 8 p] = itlep X ef - ifi C Cr,A&p Q
X E=) ,
[fif,
= 3 T efghRh ,
241 = ifi I$
Introducing
Cf&Y_‘),
(56)-(59)
(Y-l),
&+A,(&+&r)].
(63)
projected along the BF frame axes, the compoare then (64)
or
(56) ,
q
(55)
where (25) has been used. Using the transformation equations (20), (22), and (23) with the help of the above results, it is now easy to verify that ii’+-, &I
Cf+
(54)
(@=x2).
= -C’&
(57)
+ @I
cP erapAp,
(65)
where (55) has been used. Here (64) comes from the fact that fif does not commute with Cfa as seen from (55). We see that the only source of angular momentum along the z axis of the BF frame is the z components of the electronic and so-called vibrational angular momentum (as it ought to be). If p4 is chosen to be the determinant Y’ and [&l_c] = 0, then th e f orms (64) and (65) persist for the q-space.
into (5) gives 5. Molecular
(60) where the electronic angular momentum e, and socalled vibrational angular momentum c(I are defined by [6]
(61)
translational-rovibronic
hamiltonian
Now we substitute for g from (50) and for the elementsg’O from table 1 in the generalized hamiltonian (2) In the first place, as in the non-linear case, this again appears to be a complicated task (which it likewise is!). However, we may have similarly a remarkable simplification if: (i) pq in (2) is chosen to be the determinant * Y’; in turn (2), after expansion, takes the analogous form to that for the non-linear case:
and
(62) The first term in (60) is the angular momentum of the center of mass of the molecule with respect to the origin of the LF frame, and the second term is the
* pq may be taken minus the determinant Y’ in the volume elements_ However, this does not produce any change in the present calculation.
R Islampour.
M. Kasha / 7hnslational-rovibronic
hamiltonian
163
II
6. Conclusion
where (50) has been used. (ii) (65) is expressed in terms of the angular momentum operators defined by (61), (62) and (65). (iii) [cm, ~1 = 0, which it actually is [6]. Following the above remarks, and recognizing that p is only a function of the Q and both Y’ and A, are only functions of rotational variables, the molecular translational-rovibronic hamiltonian for the linear molecule is derived to be
Appendix:
A= -tw/2M +
+
( (-fn’
-12
In this paper our previous derivation of the generalized coordinate complete (non-relativistic) molecular hamiltonian for non-linear molecules is extended to the linear molecule case. The derivation is parallel to the previous one, with the initial difference of course being the simplified molecular coordinate disposition along the z auk. In previous work, the derivation of the complete rotation-vibration-electronic hamiltonian has been given for the linear case of diatomic molecules [7- 111. The generai conclusions presented in our previous paper 111 apply equally well to both non-linear and linear molecules, since the two derivations are so analogous.
c v?/21,1- A2c St
S
F
a2//aQi
v,
-
VI /21?1 N
)
)
The matrix
Y-’
For the eulerian angles 0,@, x the matrix X-t defined in the appendi__ of 1 IO be
[sin x x-l=-
I
-csc 0 cos x
cosx
csc 0 sin x
Lo
is
cot 0 cosq -cot
0
0 sin x f. I
(A-1)
J
Then (34) gives Y-‘=-fcosxKsin x
-csc cscOsin)i e cos x 0
1 0 where (Y,0 =x,J’; r= {r,,} denotes the set of electronic variables and Q = {Qk} the set of vibrational variables. The term I/ is much simpler than for the non-linear case [l], and it is absorbed in the fourth term of (67). In (67) the first three terms represent the translational, electronic and vibrational contributions to the total kinetic energy of the molecule, respectively_ The fourth term is the contribution of rotational kinetic energy and the energy of interactions (between vibrations and rotations plus electronic motions) to the total kinetic energy of the molecule_ The coordinatedependent part of the kinetic energy U is also absorbed in the fourth term. Finally, the last term is the potential energy of the entire system of electrons and nuclei.
Therefore,
(y’)-‘=
the determinant
-
0 1 01 1.
(A-2)
Y’ is
Isinx
(A-3)
cos x
or Y’ = -sin
8.
References [ 11 R. Islampour and 51.Esshs. Chem. Pi]\-s. 74 (1983) 67. [1] 9. Pod&l+-,
Phys. Rev_ 32 (1928)
8 12.
164
R. Islampour. M. Kasha / Translational-rovibronic
131 EC. Kemble,The fundamental principles of quantum mechanics (Dover. New York, 1937. 1958) p. 227. [4] A. Sayvetz. J. Chem. Phys. 7 (1939) 383. 15 I C. Eckart, Phys. Rev. 47 (1935) 552. [6] J.K.C. Watson. Mol. Phys. 19 (1970) 465. 171 R. de L. Kronig. Band spectra and molecular structure (Cambridge Univ. Press, London, 1930) pp. 6-16.
itami&onian 11
[S] J.H. van Vleck, Phys. Rev. 33 (1929) 467. [Y] G.H. Diekc, Phys. Rev. 47 (1935) 870. [IO] P.R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, 1979) p. 129. [l l] R. T Pack and J.O. Hirschfelder, J. Chcm. Phys. 49 (1968) 4009.