A. 6. Sannigrahl and Ranian Das' Indian Institute of Technology Kharagpur-721302,India
II Selection Simple Derivation of Some Basic Rules
According to the quantum mechanical theory of interaction of matter and radiation2 an electric-dipole transition between two stationary states, IJ.1) and I$z) of a system can occur only when a t least one component of the transition moment, ($11 il$p) in nonzero. This conclusion is based on several approximations the most important of which are (1) the radiation field is weak, and (2) the dimension of the interacting system (atoms or molecules) is so small compared to the wavelengths of the radiation that the radiation field over the space occupied by the system may be assumed to he constant. The conditions which must he met in order to obtain nonzero transition moments are called "selection rules." Thus a selection rule tells us whether a given transition will be allowed. The stationary states of simple systems like a harmonic oscillator, the hydrogen atom etc., are characterized by a set of quantum numbers corresponding to each constant of the motion of the system. In these cases, it is customary to express the selection rule by stating how these quantum numbers must change in a transition so that the transition moment does not vanish. The standard textbook technique for the derivation of selection rules consists of finding analytically the relation that f 0. This must exist between 1$1) and 1$2) SO that ($11 approach requires a thorough knowledge of the wavefunctions and a number of complicated recursion relations among them. Consequently, a lot of tedious mathematical manipulation is involved in this approach. The derivation of the selection rule for the principal quantum number of the hydrogen atom, for example, is so complicated that it is not included even in many advanced textbooks on quantum mechanics. There is, however, an alternative approach due to Dirac3 which is considerably simple but not quite familiar to the chemists. In this method one starts with the assumption that the transition moment is nonzero. Then subject to the validity of this assumption, searches for a subsidiary condition that must exist between the specific (energy, angular momentum Now at T.I.F.R., Bombay, India. Pilar, F. L., "ElementaryQuantum Chemistry," McGraw-Hill Book Company, New York, 1968, Ch. 5. 3 Dirac, P. A. M., "Principles of Quantum Mechanics," oxford University Press, Calcutta, 1958, Ch. VI.
786 1 Journal of Chemical Education
etc.) eigenvalues of the two stationary states under consideration. No formal knowledge of analytical forms of the wavefunctions is needed in this technique. Because of its notahle simplicity we consider i t worthwhile to present here a didactical exposition of Dirac's idea. With this objective, we have derived the selection rules for all the four quantum numbers of the hydrogen atom and for a linear harmonic oscillator. Theory
The central idea of the Dirac method is as follows. Let & and
P be two commuting Hermitian operators correspnnding to
two constanis of the motion of a system, and let Id , ) and d2) be their simultaneous eigenfunctions in any two arbit;& stationary states. Since the condition, ($1161$2) f 0, where 6 is i,9 or f, determines the selection rule, we seek to find an algebraic equation which connects &, 8, and 6 and is a t the same time linear in 6. Obviously, a general equation of this type can be written as
z f
fd;)
B fj@)gjtP) = o
(1)
,
where fi(&) etc. denotes a function of &. Taking the matrix element of eqn. (1) between and we get
W a ) =aal$r) Bl$k=bal$a) f(&)l!bk)=f(aa)l*a) Since ($116l$z) should not he zero for an allowed transition between 1$1) and I$z), we must have
XI; [fi(adgdbd fjfaz)gj(bz)l = 0 I'
(3)
L
which determines the selection rules in terms of the eigenvalues. As these eigenvalues are assumed to be quantized we can easily obtain the selection rules in terms of pertinent
quantum numbers. Equations (1)-(3) being of avery general nature look rather clumsy; but in actual applications they take very simple forms. The crux of the whole problem therefore lies in finding a relation like eqn. (1).As we shall show, such a relation can he found easily from suitable commutation relations.
Making use of the above commutation relations and the identity,
t,a+L,.j+L,t=o
Selecllon Rules for the Hydrogen Atom
The enerev. the orbital aneular momentum. the z-comvou . . nent of the orbital angular m k e n t u m , and the z-component of the spin angular momentum of each electronic state of the hydrogen atom are quantized in terms of the quantum numThe corresponding eigenbers n .. 1.. mr. .. and m,,-. resoectivelv. . value equations are given by
we get [L2, [L2,i]] = 2h2 ( t 2 i
+ 22%)
or
L4i - 2L2it2+ it4- 2h2 ( t 2 +it2) =0
(13)
Taking thematrix element of eqn. (13) between IIC.1) and l$z), we obtain
+ 12 + 2) (11+ 12) (11 - 12 + 1) (1, - 12 - 1) = 0 114) since ($112 I$%) z 0. T o have a transition between /$I) and (11
where for the sake of brevity I $) is written for I $nlm m, ), K = 2r2&z 2 e 4 h 2and h = hI2s. I t is well-known that L2, and S, form a set of commuting operators. Since S, is not a function of space coordinates, we have
A, e,
IS,, Dl = 0
(5)
which has the same form as that of eqn. (1).Taking the matrix element of eqn. (5) between 1$1) and 1$2), we get (511101512) (m.,
- ma,) h = 0
which implies that the selection rule for m, is Am. = 0. T o determine the selection rule for ml, we shall make use of the following commutation relations: [L,, t ] = i h j
1$2). one of the four factors in eqn. (14) must be zero. Since 1's are all positive, 11 12 2 f 0, and ll 12 = 0, only when 11 = 12 = 0.But this implies that mr, = mr, = 0.For suchvalues of ml, and mr,, i t can be proved3 from eqns. (6)-(8) and analand Ly that ($lldl$2) = 0. Thus we ogous ones for have
+ +
e,
(1, - 12
(7)
[L,, i ] = 0
(8)
The corresponding relations for L, and Ly can be written a t once by making use of the cyclic permutation of x, y, and z. From eqn. (a), it can be deduced easily that the selection rule for the transition polarized in the z-direction is Amr = 0. Proceeding in the like manner with eqns. (6) and (7) we get the selection rule, Aml = f 1for the absorption of circularly polarized light, i.e., light polarized along the x or y axis. If one is interested in getting the selection rule for the x- and y component separately, then the following simple trick will do. From eqns (6) and (71, we get
+ 1) (1, - l2 - 1) = 0
which gives the selection rule, A1 = f 1.The same selection rule holds for the x- and y-component as well. Lastly, we derive the selection rule for n . Since [L,, i] = 0, we have
[R,[L,, ill
(6)
[t,, 91 = -ih4
+
( $ , ~ R t , i- Rit,
- t,iR+ ii,Rl$d = h ( $ ~ l i l $ z )(E~mr,- Elmr, - Ezmi, +Ezmr,) =
h(Jillil$d (E1rE2) (mr,-ml,) = 0
(16)
Since ($112($2) z 0 and mr, = mra, we conclude that ( E l E2) can take any arbitrary value other than zero. Thus we have the selection rule, An = anything (excluding zero). For the x-component of the transition moment, we have from eqn. (9)
[R, [L, [L, 1111= IR, h r l Proceeding in the usual manner, we get ( $ ~ l a l $ d(El - E d (mr, - mh - 1) (mr, - mr2
which has the same form as that of eqn. (1). The matrix element of eqn. (9)with respect to and 1$2) is given by
(15)
=0
Expanding eqn. (15) and taking the matrix element between 111.1) and I$d, weget
(17)
+ 1) h2 = 0
(18)
which also leads to the selection rule, An = anything (excluding zero), since Am1 = f 1. The selection rule for the y-component is the same as that for the z- and the x-component and can be derived in the like manner. Selection Rule for a Linear Harmonic Oscillator
*
which leads to the selection rule, Am1 = 1.A similar relation holds good for the y-component. The selection rule for 1 can be obtained from the following commutation relations:
T o derive the selection mle for a linear harmonic oscillator, it is convenient to work in the dimensionless repre~entation.~ In this representation
A n d ~ r s o n.I.. M.. "lnrrodurtion t o Quantum Chemistry," W. A. Benjsmm, lnr.. Y e w York. 1959,Ch. 5.
Volume 57, Number 11, November 1980 1 787
1 = -2i( ( h , l $ w + 1 ) - (h,lfiO2-1)
where m is the mass of the oscillator, v is its classical frequency, and u is the vibrational quantum numher. Defining a pair of adjoint operators by ~*=@+i8
(20)
and making use of the commutation relation, [fl, p] = i, it can he shown in a straightforward manner that
A=
(f+f-
+ 1)/2 = ( i - f + - 1)/2
[A, i*] =
+ fa
Since, fl = (f+ - P-)/2i a n d P = (i+ eqn. (22) [A, 2 1 = (f+
(21) (22)
+ i-)/2, we have from
+ y_)/2i = -i@
(23)
and [A, PI = (f+ - f_)Iz = i 8
(24)
Therefore, [A, [A, 8 11 = -i[A,
PI = 8
(25)
Expanding eqn. (25) and taking the matrix element between and IJ..,), weget ($,,IA28
- 2 A 8 A + llRz - 21$",) = o
are orthogonal, (J..,Ifll J.., z 0, only when Au = * Simple 1. symmetry arguments may also be used in this case
Since $,'s
to predict whether an electric-dipole transition is allowed or forbidden, As the transition moment is a physical quantity, i t must remain invariant with respect to any symmetry operation on the system. This operation implies that the inteXJ."?must he totally symmetric. In the present case grand, J.", J."" with u = even and odd are, respectively, symmetric and antisymmetric with regard to the origin, and f (consequently 2 ) is antisymmetric. Therefore, to have a totally symmetric integrand, we must have J.,, = symmetric and J.,,= antisymmetric or vice versa. Denoting the symmetric and antisymmetric functions by g and u, respectively, we have the following selection rule for a linear harmonic oscillator:
-
u-g ueu g-g The preceding exact treatment, however, shows that all u g transitions are not allowed. Since the hydrogenic wavefunctions with 1 = even and odd are symmetric and antisymmetric, respectively with regard to the origin, i t can be similarly deduced that transitions are allowed between states having even and odd values of I. Symmetry arguments prove especially useful in the derivation of selection rules for the electronic and vibrational transitions in molecules possessing a certain degree of symmetry. Concluding Remarks
which implies that (J.,,IxJJ.,,) Z 0, only when Au = f 1. Thus the selection rule for a linear harmonic oscillator is Au =
* 1.
The above selection rule can also be derived quite easily by noting that i+and F- act as a step-up and a step-down operator, respectively, with regard to the energy eigenvalues of a linear harmonic oscillator. Thus we have
The transition moment integral, (J.,,lfl(J.uz) now becomes
788 1 Journal of Chemical Education
The derivation of some hasic selection rules as presented in this article should deserve special mention in an elementary course of quantum chemistry. While using this ap~roach,one needs to memorize practically nothing. The basic commutation relations can he derived readily once the expressions for the relevant operators are known. The present method can he applied to any constant of the motion of a system which can he characterized by a quantum number. For example, the component of the angular momentum along the internuclear axis of alinear molecule is aconstant of the motion. The corresponding eigenvalue equation is L, I J . ) = Ah1 J.), where A is a quantum number similar to the mi quantum number of the hydrogen atom. This similarity indicates that the selection rule for A is AX = 0, f 1.The selection rule for L, ML, etc., of a many-electron atom can he derived as in the case of the hydrogen atom, provided its Hamiltonian is approximated by a central-field Hamiltonian.
