A. B. Sannigrahi Indian Institute of Technology Khorogpur 2, lndio
Proiection Operators and Their Simple Applications in Chemistry
Quantum chemistry and group theory today form an integral part of many undergraduate and almost all graduate cumcula in chemistrv. A maior part of the conce~tual framework of these s ~ b j ~ cist sbased on the axioms oi linear aleebra and the concept of operators. Most frequently in quantum chemistry, one is required to find out ;he elgenvertors and associated eigenvalues of a hermitian operator. These eigenvectors are usually expressed as linear combinations of a finite set of basis vectors. Construction of molecular orhitals as linear combinations of atomic orbitals (LCAO-MO) is a familiar example of this type. These molecular orbitals are determined by diagonalizing the matrix of the Hamiltonian operator defined with respect to the basis set of atomic orbitals. In most cases, the order of ~- this ~ - matrix -~~~ ~ ~is too hieh to he solved without the help of a digital computer. ~ ; tthe problem can he greatly simplified, if the molecule under consideration possesses some symmetry (note that the symmetry of the molecule is possessed by its molecular orbitals also). Under such conditions, i t is very convenient to make use of a suitable projection operator which is capable of projecting, from s o m e a r b i t r a j function, any function i f desired symmetry. In fact, whenever there is some symmetry in a system, and one is required to construct its symmetryadapted functions like molecular orbitals, normal coordinates of vibration etc., the projection operator method proves extremely useful. Most of the applications of projection operators in chemistry have been inspired by the outstanding works of Lowdin (1-4). Besides these original sources (1-4) which are somewhat difficult to follow because of their abstract nature, a fairly good coverage of projection operators and their applications in chemistry may also be found in many textbooks (5-8). With the possible exception of (5), these treatments, however, seem to he casual. Since many important problems in quantum chemistry can be simplified to a great extent by the projection operator formalism, we feel that this method should deserve more attention than what is usually paid to i t in elementary texts. I t is, therefore, the purpose of the present article to examine the properties of projection operators .from the viewpoint of their applicability in chemistry in sufficient detail so that a student need not he mechanical while applying them to actual problems. Several simple applications have been included to illustrate the mathematical formalism developed. For more advanced applications, an interested reader should consult the articles bv Lowdin (3, 4 ) and references cited therein. Attempts have been made to make the presentation simple. It is hoped that some formal acquaintance with linear algebra, group theory, and eigenvalue problems in quantum chemistry would enable the student to follow the article. Let us consider the expansion =
2 i-1
a&
a, =
I$(* fdr
(2)
where the limits of integration are the same as the expansion interval in eqn. (1). Using Dirac's notation (in which mi = (4, >, mi* = < 6il, mi@ = Mi > < $11, and S&*@,dr = < 4iJ4j> ) eqn. (1) may be written as
Operating both sides of eqn. (3) by I q5j > < 6, I one ohtains
Thus the mathematical construct I 41 > < 41 1 ,known as a dyad acting on f selects its jth component a j 1 4j >. For geometrical reasons (given below) this construct is called a projection operator. Replacing functions by vectors we may write eqn. (1) as
-
c = 1a+j
(5)
where V is an arbitrary vector in n-dimensional space spanned by the set of orthonormal basis vectors Zi which may be represented by the column matrix (0 0 0 . . . 10 0 . . . 0); where 1 occurs in the ith row. Since vrvj = 6u, where e, is the transpose of \i, (in case of complex vectors Ti should he replaced by ?(+,the complex conjugate transpose or adjoint of 61, we obtain from eqn. (5)
I t is apparent from eqn. (6) that Gig, has the property of projecting an arbitrary vector V onto the axis of its associated basis vector Gi. Hence Ti? and analogously I 4i -> < I are called projection operators. Denoting them as Pi, i t can he easily deduced that
mt
where F,, is a n X n unit matrix (it is unity for Pi = I @i > < 4, I). Equations (7)-(9) express the three most fundamental properties of a projection operator. From the first two of these equations, one says that projection operaton are idempotent and mutually exclusive, and the last relationship is referred to as the repolution of the identity. These relations also imply that P,_has+nz inverse. P,-1 exists, then from eqn. (7) we get, PirlPiPi = Pi-'Pi, i.e., Pi = I, which is in contradiction to eqn. (9). In case of complex vego?, we have further pit = (
