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The SO(2,1) dynamic-group approach to the rotation—vibration spectra of diatomic molecules
Volume 175, number 1,2
30 November 1990
CHEMICAL PHYSICS LETTERS
The SO ( 2,1) dynamic-group approach to the rotation-vibration spectra of diatomic molecules Dong J. Lee a,r, Kook Joe Shin b and Shoon K. Kim c a Department ofApplied Chemistry, National Fisheries Universityof Pusan, Nan&u, Pusan 608-023, Korea b Department of Chemistry, Seoul National University,Seoul 151-742, Korea ’ Department of Chemistry,Temple University,Philadeiphia. PA 19112, USA Received 2 July 1990; in final form 18September 1990
The rotation-vibration spectra of diatomic molecules are discussed with the aid of the SO ( 2.1) dynamic group. The eigenvector of the compact generator is explicitly obtained by realizing the generators in the four-dimensional polar coordinates. A special representation of the radial part of the generators allows us to describe the discrete and continuous states of the Morse oscillator,
1. Introduction
Since 1982, when Iachello and Levine [ 1 ] suggested a Lie algebraic method to study the rotation-vibration spectra of diatomic molecules called the vibron model, algebraic methods have attracted much attention and have proved to be effective in the study of molecular motions [ 2-4 ] . Since rotation-vibration spectra arise from the dipole deformation, it has been assumed that the Hamiltonian belongs to the dynamic group U(4), which can be expressed in terms of four boson creation and annihilation operators. In turn, the four boson operators are divided by the scalar and vector boson operators. It should be required that the Hamiltonian conserves the number of total bosons and is invariant with respect to rotation. Then, we may obtain the chains of subgroups of U( 4) necessary to describe the rotation-vibration spectra. The rotation-vibration spectra are expressed with the aid of the eigenvalues of the invariant operators for the subgroups. The aim of the present paper is to investigate the rotation-vibration spectra of diatomic molecules by introducing the generators of a noncompact special orthogonal group SO (2,1), which are expressed by four boson operators. Explicit realizations of the generators give us the quantum numbers to discuss the rotation-vibration spectra of diatomic molecules. It is possible that the eigenvector of the generator can be obtained by realizing the generator in the four-dimensional polar coordinates. The transformation of the radial part in the generators into a special representation allows us to express the energy of the Morse oscillator in the discrete and scattering states [ 5,6].
2. Theory Let us consider the following generators expressed in terms of the boson operators s, =t(n+-a++n.n+a+a++~~)
3 Sz=-it(nt-nt-i.x+ota+-~u),
Sj=~(~+~~tn~Ir+tu+otu~+), (1)
’ To whom correspondence should be addressed. 0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland )
87
Volume 175,number 1,2
CHEMICALPHYSICSLETTERS
30 November 1990
where nt (n) and crt (a) are the vector boson-creation (-annihilation) and scalar boson-creation (-annihilation) operators, respectively defined as n+=(a],af,
al) ) n=(a,,a&
a,),
a+=aj,
u=u4_
(2)
The boson operators satisfy the following commutation relations: [at,
41=& 9 [a?, aL]~0,
[ai, ak] =O,
i,k= 1,2,3,4 .
(3)
The above generators are those of the noncompact special orthogonal group SO (2,1), since the following commutation relations hold: [S,,S,]=-is,,
[S,,&]=iS,,
[S,,S,]=iS,.
(4)
Let A be the total-boson-number operator. Then, the compact generator S, is s,=j(A+z).
(5)
In the coordinate realization, the boson operators are expressed as
(6) where we have chosen h = 1. Substitution of eq. (6) into eq. ( 1) leads to (7) Transforming Xi into the polar coordinate (I, 0, q, C) x,=rsin8cosqsini, (O
x2=rsin0sin~sin[,
x3=rcos6sin<,
x4=rcosC
x, OGfj7,(<2x) )
(8)
reduces eq. ( 7) to
(9) where j2 is the Casimir generator p=_
-_ ’
of the SO( 3) group, which is the total angular momentum operator
a sine2 _ -’ a2 ae sin% a012*
sin 8 a0
( )
(10)
As shown in eq. (5 ), S, is related to the total boson number. Using the conventional method of separation of
variables, we may write M(r)
G(C) =2(N+2)F(r)
G(C) ,
Wa)
(lib)
CHEMICAL PHYSICS LETTERS
Volume 175, number 1,2
30 November 1990
-[-g(r$)-r2+ -$“)=Z(Nf2)F(r).
(llc)
Let the eigenvalue of .?’be J(J+ 1) . In reference to the differential equation given in eq. ( 11b), the substitution z=cosc,
G(z)=(l-z~)“~C(Z)
(12)
leads to the equation
(13) When a= w (w+ 2)) the solution exists and the result is the well-known Gegenbauer differential equation [ 7 ]
The angular momentum quantum number J takes the values J=O, I, 2, ...) w
(15)
and the quantum number w will be restricted by the total boson number, N. Via the result of eq. (14) eq. ( 11c ) becomes [$$(r$)-r2+
w(;2+2)
+2(N+2+‘(+0
(16)
_
Introduction of the following variable transformation, r2=y,
F(r) =y w/2exp( - 1~) L(y) ,
(17)
into eq. ( 16) leads to the associated Laguerre differential equation d2 ydy2 + (w+2-y)
$ +v Liti+‘)( >
(18)
)
where u is 0 or a positive integer and defined as (19)
v=f(N-w).
The quantum number o is N, N-2, .... 1 or 0. Later, it will be shown that u is the vibrational quantum number for the Morse oscillator. The number of total bosons is related to the maximum number of the vibrational states as N=2v,,,
or
2v,,, + 1 , if N is an even or odd integer.
(20)
Let the eigenstate of S, be 1NWJM). Then, the normalized eigenstate in the position representation is
INc~At4)=(-1)~
2u(0+ 1 )(2J+ ’)“!(0-J)!(J)2(Jx2(v+w+l)!(w+J+l)!(J+~M()!
i”1 )! “2F$(r)
GJ (,+‘I yM(() p) w J 3 ,
(21)
where F:(r) =P exp( - tr’) LW+’ ” (r) , G-‘,(O=sin’~CJ;t_‘Jcos c) , Y,M(0,p)=P~MMI(cosf3) exp(iA4q).
(22)
L;+’ (r*) and C’,‘_$(cos 0) are the associated Laguerre polynomial and Gegenbauer polynomial, respectively. The eigenvalue of the Casimir operator of SO(2,l) is
Here,
89
Volume 175, number 1,2
C~[SO(Z,l)]
CHEMICAL PHYSICS LETTERS
IN~JM)=-(S:tSt-S:)INwJM}=aw(w+2)1NwJM),
30 November 1990
(23)
In order to prove that v is the vibrational quantum number of the Morse oscillator, let us consider the radial part in the generators [ $61. Transforming F(r)=(2k)-‘exp(-<)
CD(C), r=kexp(-<),
(24)
we may have S~0(<)=(2k)-2exp(25)
(
S$@(~5=-(2k)-~exp(20
$+k4exp(-45)-(w+1)2
)
$-k4exp(-4&)-(wtl)’
@i(r),
>
Q(r),
(25)
where k is a parameter and SY= (2k) exp( -0
Si(2k)-’ exp(<) .
(26)
Let us consider the following equation, [(+64A)Sf-(f+64A)S$+8B]0(~)=0.
(27)
Introducing an arbitrary tilting angle cr [ 5,6], we have exp( -is&r)
[(f-64A)SF-(~+64A)S~t8R]
exp(iS$a) &(r)=O,
(28)
where &5)=exp(
-iS$a)
Q(C) exp(iS5c).
(29)
With the aid of the commutation relations between the generators, we have [-64A(SftS$)
exp(a)+f(Sf-S$)
exp(-o)+8B]@5)=0.
(30)
Substitution of eq. (24) into eq. (29) leads to
(
~-8A(2k)4exp(2~)exp(-4e)+8~(2k)‘exp(a)exp(-2e)-(rrl+l)2
(31)
Let us identify (2k)2=exp( -a),
2+R.
(32)
Then, we may obtain -2,4exp(-2R)+2Bexp(-R)-i(w+l)2 The eigenvalue of the Hamiltonian for the Morse oscillator
(33) is
where the mass and constant of the Morse oscillator have been taken unity. It should be noted that the ei90
CHEMICAL PHYSICS LETTERS
Volume175, number 1.2
30 November 1990
genvalue of the Morse potential is shifted from that of the Casimir operator of the SO(2,l) dynamic group, that is, E,= -f{C~[SO(Z,l)]+t}.
(35)
This means that symmetry breaking occurs, when the Morse potential is dealt with by a group containing the subgroup SO (2,1). The angular momentum operator I2 in eq. ( 10) describes the rotational energy. Thus, combining the rotational energy with E, of eq. (34), we can express the discrete rotation-vibration energy state of diatomic molecules as E,=E,-Q(o+l)*+
&(Jfl)
I
0
where r. is the equilibrium distance of the oscillator. The result is just the same as the case of the U (4) I SO (4) r) SO( 3) chain for the rotation-vibration spectra [ 11. The scattering state of the Morse oscillator corresponds to the continuous values of or)in the principal series of representation, that is, D,(w) , + (co+ l)=iz
(A is real) .
(37)
Thus, the vibrational energy in the scattering state is &=‘A2 2 -
(38)
3. Conclusion We have discussed the rotation-vibration spectra of diatomic molecules with the aid of the SO (2,1) dynamic group. The realization of the generators for the vibron model in the four-dimensional space gives us the explicit representation of the eigenvector of the compact generator. The transformation of the radial part in the generators into a special representation allows us to obtain the discrete and continuous states of the Morse oscillator. The application of the interesting dynamic group SO(4,2) [ 51 to the vibron model of diatomic molecules is in progress. The present result is valuable in the study of the rotation-vibration spectra by the SO( 4,2) group.
Acknowledgement This work was supported by the Korean Science and Engineering Foundation, 1989.
References [ I] F. Iachello and R.D. Levine, J. Chem. Phys. 77 (1982) 3006. [ 21 OS. van Roosmalen, F. Iachello, R.D. Levine and A.E.L. Dieperink, J. Chem. Phys. 79 ( 1983) 2515. [3] OS. van Roosmalen, 1. Benjamin and R.D. Levine, J. Chem. Phys. 8 I (1984) 5986. [4] S.K. Kim, I.L. Cooper andR.D. Levine, Chem. Phys. 106 (1986) 1. [ 51 B. Wybourne, Classical groups for physicists (Wiley, New York, I974 ). [6] A.O. Bantt, A. Inomata and R. Wilson, J. Math. Phys. 28 (1987) 605. [ 7 ] I.S. Gradshteyn and 1.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1980).
91
30 November 1990
CHEMICAL PHYSICS LETTERS
The SO ( 2,1) dynamic-group approach to the rotation-vibration spectra of diatomic molecules Dong J. Lee a,r, Kook Joe Shin b and Shoon K. Kim c a Department ofApplied Chemistry, National Fisheries Universityof Pusan, Nan&u, Pusan 608-023, Korea b Department of Chemistry, Seoul National University,Seoul 151-742, Korea ’ Department of Chemistry,Temple University,Philadeiphia. PA 19112, USA Received 2 July 1990; in final form 18September 1990
The rotation-vibration spectra of diatomic molecules are discussed with the aid of the SO ( 2.1) dynamic group. The eigenvector of the compact generator is explicitly obtained by realizing the generators in the four-dimensional polar coordinates. A special representation of the radial part of the generators allows us to describe the discrete and continuous states of the Morse oscillator,
1. Introduction
Since 1982, when Iachello and Levine [ 1 ] suggested a Lie algebraic method to study the rotation-vibration spectra of diatomic molecules called the vibron model, algebraic methods have attracted much attention and have proved to be effective in the study of molecular motions [ 2-4 ] . Since rotation-vibration spectra arise from the dipole deformation, it has been assumed that the Hamiltonian belongs to the dynamic group U(4), which can be expressed in terms of four boson creation and annihilation operators. In turn, the four boson operators are divided by the scalar and vector boson operators. It should be required that the Hamiltonian conserves the number of total bosons and is invariant with respect to rotation. Then, we may obtain the chains of subgroups of U( 4) necessary to describe the rotation-vibration spectra. The rotation-vibration spectra are expressed with the aid of the eigenvalues of the invariant operators for the subgroups. The aim of the present paper is to investigate the rotation-vibration spectra of diatomic molecules by introducing the generators of a noncompact special orthogonal group SO (2,1), which are expressed by four boson operators. Explicit realizations of the generators give us the quantum numbers to discuss the rotation-vibration spectra of diatomic molecules. It is possible that the eigenvector of the generator can be obtained by realizing the generator in the four-dimensional polar coordinates. The transformation of the radial part in the generators into a special representation allows us to express the energy of the Morse oscillator in the discrete and scattering states [ 5,6].
2. Theory Let us consider the following generators expressed in terms of the boson operators s, =t(n+-a++n.n+a+a++~~)
3 Sz=-it(nt-nt-i.x+ota+-~u),
Sj=~(~+~~tn~Ir+tu+otu~+), (1)
’ To whom correspondence should be addressed. 0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland )
87
Volume 175,number 1,2
CHEMICALPHYSICSLETTERS
30 November 1990
where nt (n) and crt (a) are the vector boson-creation (-annihilation) and scalar boson-creation (-annihilation) operators, respectively defined as n+=(a],af,
al) ) n=(a,,a&
a,),
a+=aj,
u=u4_
(2)
The boson operators satisfy the following commutation relations: [at,
41=& 9 [a?, aL]~0,
[ai, ak] =O,
i,k= 1,2,3,4 .
(3)
The above generators are those of the noncompact special orthogonal group SO (2,1), since the following commutation relations hold: [S,,S,]=-is,,
[S,,&]=iS,,
[S,,S,]=iS,.
(4)
Let A be the total-boson-number operator. Then, the compact generator S, is s,=j(A+z).
(5)
In the coordinate realization, the boson operators are expressed as
(6) where we have chosen h = 1. Substitution of eq. (6) into eq. ( 1) leads to (7) Transforming Xi into the polar coordinate (I, 0, q, C) x,=rsin8cosqsini, (O
x2=rsin0sin~sin[,
x3=rcos6sin<,
x4=rcosC
x, OGfj7,(<2x) )
(8)
reduces eq. ( 7) to
(9) where j2 is the Casimir generator p=_
-_ ’
of the SO( 3) group, which is the total angular momentum operator
a sine2 _ -’ a2 ae sin% a012*
sin 8 a0
( )
(10)
As shown in eq. (5 ), S, is related to the total boson number. Using the conventional method of separation of
variables, we may write M(r)
G(C) =2(N+2)F(r)
G(C) ,
Wa)
(lib)
CHEMICAL PHYSICS LETTERS
Volume 175, number 1,2
30 November 1990
-[-g(r$)-r2+ -$“)=Z(Nf2)F(r).
(llc)
Let the eigenvalue of .?’be J(J+ 1) . In reference to the differential equation given in eq. ( 11b), the substitution z=cosc,
G(z)=(l-z~)“~C(Z)
(12)
leads to the equation
(13) When a= w (w+ 2)) the solution exists and the result is the well-known Gegenbauer differential equation [ 7 ]
The angular momentum quantum number J takes the values J=O, I, 2, ...) w
(15)
and the quantum number w will be restricted by the total boson number, N. Via the result of eq. (14) eq. ( 11c ) becomes [$$(r$)-r2+
w(;2+2)
+2(N+2+‘(+0
(16)
_
Introduction of the following variable transformation, r2=y,
F(r) =y w/2exp( - 1~) L(y) ,
(17)
into eq. ( 16) leads to the associated Laguerre differential equation d2 ydy2 + (w+2-y)
$ +v Liti+‘)( >
(18)
)
where u is 0 or a positive integer and defined as (19)
v=f(N-w).
The quantum number o is N, N-2, .... 1 or 0. Later, it will be shown that u is the vibrational quantum number for the Morse oscillator. The number of total bosons is related to the maximum number of the vibrational states as N=2v,,,
or
2v,,, + 1 , if N is an even or odd integer.
(20)
Let the eigenstate of S, be 1NWJM). Then, the normalized eigenstate in the position representation is
INc~At4)=(-1)~
2u(0+ 1 )(2J+ ’)“!(0-J)!(J)2(Jx2(v+w+l)!(w+J+l)!(J+~M()!
i”1 )! “2F$(r)
GJ (,+‘I yM(() p) w J 3 ,
(21)
where F:(r) =P exp( - tr’) LW+’ ” (r) , G-‘,(O=sin’~CJ;t_‘Jcos c) , Y,M(0,p)=P~MMI(cosf3) exp(iA4q).
(22)
L;+’ (r*) and C’,‘_$(cos 0) are the associated Laguerre polynomial and Gegenbauer polynomial, respectively. The eigenvalue of the Casimir operator of SO(2,l) is
Here,
89
Volume 175, number 1,2
C~[SO(Z,l)]
CHEMICAL PHYSICS LETTERS
IN~JM)=-(S:tSt-S:)INwJM}=aw(w+2)1NwJM),
30 November 1990
(23)
In order to prove that v is the vibrational quantum number of the Morse oscillator, let us consider the radial part in the generators [ $61. Transforming F(r)=(2k)-‘exp(-<)
CD(C), r=kexp(-<),
(24)
we may have S~0(<)=(2k)-2exp(25)
(
S$@(~5=-(2k)-~exp(20
$+k4exp(-45)-(w+1)2
)
$-k4exp(-4&)-(wtl)’
@i(r),
>
Q(r),
(25)
where k is a parameter and SY= (2k) exp( -0
Si(2k)-’ exp(<) .
(26)
Let us consider the following equation, [(+64A)Sf-(f+64A)S$+8B]0(~)=0.
(27)
Introducing an arbitrary tilting angle cr [ 5,6], we have exp( -is&r)
[(f-64A)SF-(~+64A)S~t8R]
exp(iS$a) &(r)=O,
(28)
where &5)=exp(
-iS$a)
Q(C) exp(iS5c).
(29)
With the aid of the commutation relations between the generators, we have [-64A(SftS$)
exp(a)+f(Sf-S$)
exp(-o)+8B]@5)=0.
(30)
Substitution of eq. (24) into eq. (29) leads to
(
~-8A(2k)4exp(2~)exp(-4e)+8~(2k)‘exp(a)exp(-2e)-(rrl+l)2
(31)
Let us identify (2k)2=exp( -a),
2+R.
(32)
Then, we may obtain -2,4exp(-2R)+2Bexp(-R)-i(w+l)2 The eigenvalue of the Hamiltonian for the Morse oscillator
(33) is
where the mass and constant of the Morse oscillator have been taken unity. It should be noted that the ei90
CHEMICAL PHYSICS LETTERS
Volume175, number 1.2
30 November 1990
genvalue of the Morse potential is shifted from that of the Casimir operator of the SO(2,l) dynamic group, that is, E,= -f{C~[SO(Z,l)]+t}.
(35)
This means that symmetry breaking occurs, when the Morse potential is dealt with by a group containing the subgroup SO (2,1). The angular momentum operator I2 in eq. ( 10) describes the rotational energy. Thus, combining the rotational energy with E, of eq. (34), we can express the discrete rotation-vibration energy state of diatomic molecules as E,=E,-Q(o+l)*+
&(Jfl)
I
0
where r. is the equilibrium distance of the oscillator. The result is just the same as the case of the U (4) I SO (4) r) SO( 3) chain for the rotation-vibration spectra [ 11. The scattering state of the Morse oscillator corresponds to the continuous values of or)in the principal series of representation, that is, D,(w) , + (co+ l)=iz
(A is real) .
(37)
Thus, the vibrational energy in the scattering state is &=‘A2 2 -
(38)
3. Conclusion We have discussed the rotation-vibration spectra of diatomic molecules with the aid of the SO (2,1) dynamic group. The realization of the generators for the vibron model in the four-dimensional space gives us the explicit representation of the eigenvector of the compact generator. The transformation of the radial part in the generators into a special representation allows us to obtain the discrete and continuous states of the Morse oscillator. The application of the interesting dynamic group SO(4,2) [ 51 to the vibron model of diatomic molecules is in progress. The present result is valuable in the study of the rotation-vibration spectra by the SO( 4,2) group.
Acknowledgement This work was supported by the Korean Science and Engineering Foundation, 1989.
References [ I] F. Iachello and R.D. Levine, J. Chem. Phys. 77 (1982) 3006. [ 21 OS. van Roosmalen, F. Iachello, R.D. Levine and A.E.L. Dieperink, J. Chem. Phys. 79 ( 1983) 2515. [3] OS. van Roosmalen, 1. Benjamin and R.D. Levine, J. Chem. Phys. 8 I (1984) 5986. [4] S.K. Kim, I.L. Cooper andR.D. Levine, Chem. Phys. 106 (1986) 1. [ 51 B. Wybourne, Classical groups for physicists (Wiley, New York, I974 ). [6] A.O. Bantt, A. Inomata and R. Wilson, J. Math. Phys. 28 (1987) 605. [ 7 ] I.S. Gradshteyn and 1.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1980).
91