- I am trying to calculate the polarizability, first hyperpolarizability and second hyperpolarizability values from Gaussian output files. The calculated values of polarizability in two different basis set methods are in negative. What does it mean for the nagative value of polarizabilty we obtained?
Polarizabilities for real positive frequencies have poles at excitation energies. Close to the pole from the left side the polarizability goes to plus nfinity, and close to the the pole from the right side to minus infinity. Therefore, frequency-dependent polarizabilities can be negative for frequencies larger than the first excitation energy, even for the ground state.
Negative polarizability for excited states can be understood by considering the expression for this quantity, involving a sum over all states barring the state one is considering of a quantity whose denominator consists of the difference of the energy of the state one is considering and that of the state corresponding to the summation index; the nominator is the square of the absolute value of the matrix element of the dipole moment operator with respect to these states. For excited states the latter energy difference can take both positive and negative values, implying that the sign of the associated polarizability can be both positive and negative (depending on the various energy differences and the associated dipole moments). For some relevant details, you may wish to consult the following publication (see in particular Eqs (7) and (8) herein):
http://scitation.aip.org/content/aip/journal/jcp/115/21/10.1063/1.1415085
Polarizabilities for real positive frequencies have poles at excitation energies. Close to the pole from the left side the polarizability goes to plus nfinity, and close to the the pole from the right side to minus infinity. Therefore, frequency-dependent polarizabilities can be negative for frequencies larger than the first excitation energy, even for the ground state.
Behnam and Tatiana are both correct. I would add only that the DC polarizability of a nondegenerate ground state must be positive, as can be shown by a simple exercise in 2nd order perturbation theory. There are many examples of excited states which have negstive DC polarizabilities.
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