Electron symmetry operator

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

The subsets of d orbitals in Fig. 3-4 may also be labelled according to their symmetry properties. The d ildxi y2 pair are labelled and the d yldxMyz trio as t2g. These are group-theoretical symbols describing how these functions transform under various symmetry operations. For our purposes, it is sufficient merely to recognize that the letters a ox b describe orbitally i.e. spatially) singly degenerate species, e refers to an orbital doublet and t to an orbital triplet. Lower case letters are used for one-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of... 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

For the homonuclear (HON) species, the permutation-symmetry operator had the following form Y = 83) <8) Ye S2), where 83) is a Young operator for the third-order symmetric group which permutes the nuclear coordinates and 82) is a Young operator for the second-order symmetric group which permutes the electronic coordinates. For the fermionic nuclei (H and T, spin = 1/2) the Young operators corresponded to doublet-type representations, while for the bosonic D nuclei we use operators that correspond to the totally symmetric representation. In all cases the electronic operator corresponded to a singlet representation. 

For the heteronuclear (HEN) A2B-type species, the symmetry operator was = 1(82) Ye 82)- In this case the nuclear operator was a singlet for H parrs and symmetric for D pairs. Finally for the HEN HDT-type isotopomer we had = e(82). Again in all cases the electron operators represented a singlet. For discussion of the construction of the operators, see, for example, the excellent work of Pauncz [73]. 

P is a symmetry operator ensuring the proper spatial symmetry of the function, (1,2) stands for the permutation (exchange) of both electrons coordinates and the sign in Eq. (22) determines the multiplicity for (+) Ft, represents the singlet state and for (-) the triplet state. 

In physics and chemistry there are two different forms of spatial symmetry operators the direct and the indirect. In the direct transformation, a rotation by jr/3 radians, e.g., causes all vectors to be rotated around the rotation axis by this angle with respect to the coordinate axes. The indirect transformation, on the other hand, involves rotating the coordinate axes to arrive at new components for the same vector in a new coordinate system. The latter procedure is not appropriate in dealing with the electronic factors of Born-Oppenheimer wave functions, since we do not want to have to express the nuclear positions in a new coordinate system for each operation. 

Symmetry operators leave the electronic Hamiltonian H invariant because the potential and kinetic energies are not changed if one applies such an operator R to the coordinates and momenta of all the electrons in the system. Because symmetry operations involve reflections through planes, rotations about axes, or inversions through points, the application of such an operation to a product such as H / gives the product of the operation applied to each term in the original product. Hence, one can write ... 

Because symmetry operators commute with the electronic Hamiltonian, the wavefunctions that are eigenstates of H can be labeled by the symmetry of the point group of the molecule (i.e., those operators that leave H invariant). It is for this reason that one constructs symmetry-adapted atomic basis orbitals to use in forming molecular orbitals. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

In summary, proper spin eigenfunctions must be constructed from antisymmetric (i.e., determinental) wavefunctions as demonstrated above because the total S2 and total Sz remain valid symmetry operators for many-electron systems. Doing so results in the spin-adapted wavefunctions being expressed as combinations of determinants with coefficients determined via spin angular momentum techniques as demonstrated above. In... 

One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy V is unchanged when ah of the electrons have their position vectors subjected to inversion (i r = -r). This quantum number is straightforward to determine. Because each L, S, Ml, Ms, H state discussed above consist of a few (or, in the case of configuration interaction several) symmetry adapted combinations of Slater determinant functions, the effect of the inversion operator on such a wavefunction P can be determined by ... 

The point group symmetry labels of the individual orbitals which are occupied in any determinental wave function can be used to determine the overall spatial symmetry of the determinant. When a point group symmetry operation is applied to a determinant, it acts on all of the electrons in the determinant for example, ov If... 

Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "block diagonal". That is, objects of different symmetry will not interact only interactions among those of the same symmetry need be considered. 

We now return to the symmetry analysis of orbital products. Such knowledge is important because one is routinely faced with constructing symmetry-adapted N-electron configurations that consist of products of N individual orbitals. A point-group symmetry operator S, when acting on such a product of orbitals, gives the product of S acting on each of the individual orbitals... 

In the molecular orbital theory and electronic spectroscopy we are interested in the electronic wave functions of the molecules. Since each of the symmetry operations of the point group carries the molecule into a physically equivalent configuration, any physically observable property of the molecule must remain unchanged by the symmetry operation. Energy of the molecule is one such property and the Hamiltonian must be unchanged by any symmetry operation of the point group. This is only possible if the symmetry operator has values 1. Hence, the only possible wave functions of the molecules are those which are either symmetric or antisymmetric towards the symmetry operations of the... 

Our definition of 0M applied to functions of the coordinates xx, x and x% of a point in physical space, but it can be generalized to apply to functions of any number of variables, as long as we know how those variables change under the symmetry operations. For example, if we let X stand for a complete specification of the coordinates of all the electrons (or all the nuclei) of some molecule, i.e. 

Consider a set of mutually perpendicular axes (specified by the vectors e1P e and es) with their origins at that point in the molecule which is unmoved by all the symmetry operations of the point group and let us refer the position of any nucleus to a set of axes (parallel to elt e and e3) whose origin is at the equilibrium position of the nucleus. Then the electric dipole moment of the molecule when the nuclei and electrons are in positions described by some configuration X, is the... 

The first step in the Fourier synthesis of the caffeine-pyrogallol complex was to choose an initial model so an electron density projection could be calculated. From the previous results it was determined that the unit cell was tetragonal measuring 23.26 X 23.26 X 6.99 A., and that eight caffeine-pyrogallol complex moieties reside in the unit cell. This information alone does not give any indication of the way in which the molecules are packed inside the unit cell, but the symmetry operations of the space group can be used to eliminate many models. For example, one of the symmetry operators in the unit cell is a fourfold axis. 

At this point nothing was known about the orientation of the caffeine with respect to the pyrogallol hence, as a starting point an orientation was assumed. The two components were then placed on the xy plane in a number of ways all of which were inside the limited area created by the symmetry operators of P4/n. These models were then used to predict the signs of 15 of the strongest (h,k,0) reflections. The strongest reflections were chosen because they contribute the most to the electron density. 

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