.. , |ββββ〉},

and which satisfy the following eigenvalue equation: equation(1) H^Z|m1m2m3m4〉=(m1+m2+m3+m4)ℏωH|m1m2m3m4 The symmetry-operations within the Td point group are those of one E (identity operator), eight C3 axes (proper rotations), three C2 axes (proper rotations), six S4 axes (improper rotations), and six σd planes (dihedral symmetry planes) [24]. Thus, the order of the Td group, h, is 24, and the Td point-group is isomorphic to the S4 symmetric group of permutations of four elements. It is noted that the 24 symmetry operations cannot mix states with different eigenvalues to the Zeeman Hamiltonian; that is, the matrix representations of the symmetry elements http://www.selleckchem.com/products/PD-0332991.html are block-diagonal. The function |αααα〉 is the only function with eigenvalue +2ℏωH+2ℏωH and since this function is total-symmetric it is already an irreducible representation with symmetry A  1. The four functions αααβ〉, are the only functions with eigenvalue of +ℏωH+ℏωH and these functions are therefore considered separately. The number of symmetry-adapted basis functions within each

of the irreducible representations of the Td   group is determined using Schur’s orthogonality theorems [24] and [25] that leads to equation(2) al=1h∑cg(c)χ(l)(c)∗χ(c)where al   is the number of functions with representation l  , the sum is over the classes c   of symmetry operations, g  (c  ) is the number of operations within the class, and χ(l)(c)χ(l)(c) www.selleckchem.com/products/SP600125.html and χ(c)χ(c) are the characters of the representation l   and of the set of functions MycoClean Mycoplasma Removal Kit in question, respectively. The characters χ(l)(c)χ(l)(c) are available from standard character-tables while χ(c)χ(c) is simply the number of basis functions that do not change under the

given symmetry operation. Thus, equation(3) aA1=124(1×1×4+8×1×1+3×1×0+6×1×0+6×1×2)=1 equation(4) aA2=124(1×1×4+8×1×1+3×1×0+6×(-1)×0+6×(-1)×2)=0 equation(5) aE=aT1=0aE=aT1=0 equation(6) aT2=124(1×3×4+8×0×1+3×(-1)×0+6×(-1)×0+6×1×2)=1 The four basis functions, ααβα〉, , therefore span one function with A1 symmetry and three functions with T2 symmetry (the order of the T2 symmetry is three). The full set of symmetry-adapted functions are now generated from the original set by applying the 24 symmetry operations and multiplying by the character of the symmetry operation in question as detailed elsewhere [24] and [25]. Thus, generation from |αααβ〉 gives, equation(7) Three additional functions with T  2 symmetry can be constructed in a similar manner by applying the procedure detailed in Eq. (7) to the other three functions that have an eigenvalue of +ℏωH+ℏωH, that is |ααβα〉, |αβαα〉 and |βααα〉. Finally, a basis set of functions with T2 symmetry, which consists of three orthonormal functions, can be constructed from linear combinations of the four functions generated above.