Quadratic response

An example of a quadratic response function is the first hyperpolarizability. If we are interested in $\beta_{zzz}\equiv-\langle\!\langle z;z,z\rangle\!\rangle_{0,0}$ only, we may use the following input:

**DALTON INPUT
.RUN RESPONSE
**WAVE FUNCTIONS
.HF
**RESPONSE
*QUADRATIC
.DIPLNZ
**END OF DALTON INPUT

When no frequencies are given in the input, the static value is assumed by default. If we wish to calculate dynamic hyperpolarizabilities we supply frequencies, but in this case we have two frequencies $\omega_b, \omega_c$ which are given by the keywords .BFREQ and .CFREQ (see the Reference Manual, chapter 26). The non-zero linear response functions from the operators can be generated with no additional computational costs, and all $\langle\!\langle A;B\rangle\!\rangle_{\omega_b}$ results will also be printed (in this example $\alpha_{zz}$).

The residue of a quadratic response function gives two-photon transition amplitudes. For such a calculation we supply the same extra keywords as in the linear case (Sec. 12.2.1):

**RESPONSE
*QUADRATIC
.DIPLNZ
.SINGLE RESIDUE
.ROOTS
 2 0 0 0

which in this case means the two-photon transition amplitude between the reference state and the first two excited states in the first symmetry. In general the residue of a quadratic response function corresponds to the induced transition moment of an operator $A$ due to a perturbation $B$. The $C$ operator is arbitrary and is not specified. A typical example is the dipole matrix element between a singlet and triplet state that is induced by spin-orbit coupling (phosphorescence). For this special case we have the keyword, .PHOSPHORESCENCE under *QUADRATIC, which sets $A$ to electric dipole operators and $B$ to spin-orbit operators.

The residue of a quadratic response function can be used to identify the two-photon transition amplitudes. The input below refers to the calculation of the two-photon absorption from the ground state to the first 3 excited states in point group symmetry one. In the program output the two-photon transition matrix element is given as well as the two-photon transition probability relevant for an isotropic gas or liquid. The evaluation of the transition probabilities can be done based on the transition matrix elements although they, in principle, are connected with the imaginary part of the second hyperpolarizability. The absorption cross sections are evaluated assuming a monochromatic light source that is either linearly or circularly polarized.

**RESPONSE
*QUADRATIC
.TWO-PHOTON
.ROOTS
 3 0 0 0

Another special case of a residue of the quadratic response function is the ${\cal{B}}(0\to f)$ term of magnetic circular dichroism (MCD).

**RESPONSE
*QUADRATIC
.SINGLE RESIDUE
.ROOTS
 2 2 0 0
.MCDBTERM

For each dipole-allowed excited state among those specified in .ROOTS, the .MCDBTERM keyword automatically calculates all symmetry allowed products of the single residue of the quadratic response function for $A$ corresponding to the electric dipole operator and $B$ to the angular momentum operator with the single residue of the linear response function for $C$ equal to the electric dipole operator. In other words, the mixed electric dipole--magnetic dipole two-photon transition moment for final state $f$ times the dipole one-photon moment for the same state $f$. Note that in the current implementation (for SCF and MCSCF), degeneracies between excited states may lead to numerical divergencies. The final ${\cal{B}}(0\to f)$ must be obtained from a combination of the individual components, see the original paper [17].

It is possible to construct double residues of the quadratic response function, the interpretation of which is transition moments between two excited states. Specifying .DOUBLE in the example above thus gives the matrix elements of the $z$-component of the dipole moment between all excited states specified in .ROOTS. Note that the diagonal contributions gives not the expectation value in the excited state, but rather the difference relative to the reference state expectation value.