Eigenvalues equal

An example will help illustrate these points. The px, Py and pz orbitals are eigenfunctions of the angular momentum operator with eigenvalues equal to L(L+1) h ... 

This partieular Hamiltonian, when aeting on any Slater determinant formed by plaeing N eleetrons into the SCF spin-orbitals, yields a zeroth order eigenvalue equal to the sum of the orbital energies of the spin-orbitals appearing in that determinant ... 

Since each cycle yields a single eigenvalue A = 1, the number of eigenvalues equal to one gives the number of distinct cycles. 

Thus, the functions tpi are eigenfunctions of the commutator A, S] with eigenvalues equal to zero. An operator that gives zero when applied to any member of a complete set of functions is itself zero, so that A and B commute. We have just shown that if the operators A and B have a complete set of simultaneous eigenfunctions, then A and B commute. 

The only circumstance in which a operating on an eigenvector yields the value zero is when the eigenvector corresponds to the eigenvalue A = 0, as shown in equation (4.29). Since the eigenvalue of a r]i) rj — k and this eigenvalue equals zero, we have r) k = 0 and rj must be an integer. The minimum value of A = — A is, then, zero. 

An eigencomponent routine confirms that the matrix ATA has one eigenvalue equal to zero with corresponding eigenvector [0.485, —0.485, +0.2425, —0.485, -0.485]1. It is common practice to use integers as stoichiometric coefficients. This can be achieved by dividing each component by the component of smallest modulus (0.2425), which produces the vector [2, - 2,1, - 2, - 2] corresponding to the mineral reaction... 

An example will help illustrate these points. The px, py and pz orbitals are eigenfunctions of the L2 angular momentum operator with eigenvalues equal to L(L+1) h2 = 2 h2. Since L2 and Lz commute and act on the same (angle) coordinates, they possess a complete set of simultaneous eigenfunctions. Although the px, py and pz orbitals are not eigenfunctions of Lz, they can be combined (as above to form the Gp functions) to form... 

The eigenvalues are, of course, the same as for Sz since all directions of space are equivalent. As a check, the sum of the eigenvalues equals zero, which equals the trace of S. . To find the eigenvector corresponding to A, we substitute Aj into (2.67) to get... 

The coefficient C, in the expression for ASi accounts for the asymmetry of the environment and is equal to zero when two eigenvalues are equal, or the third eigenvalue equals zero, or the last two eigenvalues equal zero. 

In all of the examples above, the bifurcation occurs when A = 0, or equivalently, when one of the eigenvalues equals zero. More generally, the saddle-node, transcritical, and pitchfork bifurcations are all examples of zero-eigenvalue bifurcations. (There are other examples, but these are the most common.) Such bifurcations always involve the collision of two or more fixed points. 

Under the C3 operation these states are again eigenstates with eigenvalues equal to the product of the individual eigenvalues, whereas operation with C2 connects jEjEj) with E iE i) and EiE j ) with lE—iEi). Forming the symmetric and antisymmetric products... 

The set Z is such a subset of the set M that the function V has degenerate critical points at points belonging to Z. The sufficient condition for a critical point to be degenerate is vanishing of the Hessian matrix determinant, i.e. vanishing of at least one eigenvalue of the Hessian matrix (the catastrophe functions in two variables have both the eigenvalues equal to zero). 

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