NONLINEAR_OPTICS
Frequency-dependent nonlinear optical response of an insulating periodic Kohn-Sham system in the independent-particle approximation (no explicit excited-state treatment, in particular no TDDFT, GW-BSE or excitonic effects). The induced polarization is expanded in powers of the electric field. With \(E(t)=\sum_ω E(ω)e^{-iωt}\), the second- through fifth-order contributions are
Here i,a,… denote Cartesian directions and ε_0 is the vacuum permittivity. Each tensor is averaged over all simultaneous permutations of its incident frequency-direction pairs. The frequency sums include ordered tuples, so no extra factorial occurs in these formulas. Negative frequencies describe complex-conjugate fields; harmonic generation uses equal positive frequencies. The input specifies photon energies ℏω_j in eV, with Ω≠0. The density matrix is expanded recursively in the fields, using all AO bands and analytic derivatives with respect to k. In the energy eigenbasis the usual population and coherence relaxation terms are
Here \(f_n\) is the equilibrium occupation. With T1_T2, \(T_1=T_2=T\) and the relaxation is \(-(ρ-ρ_{\mathrm{eq}})/T\). With separate T1 and T2, the rate for bands m and n changes smoothly with their energy difference Δε:
The width w is T1_T2_SMOOTHING_WIDTH. At exact degeneracy the rate is 1/T1 for all matrices within the degenerate band subspace. The Gaussian function of the Hamiltonian commutator gives a basis-covariant transition without band grouping. At each response step, denominators are \(ε_m-ε_n-ℏΩ-iγ_{mn}\). The current includes the field-dependent contribution required by the projected position operator. In atomic units, with r_i=i∂_{k_i}+A_i,
Let I^{(n)} denote the cell-integrated current response, including the Brillouin-zone weights and spin occupations, in SI units. The volume normalization and conversion to susceptibility are
where \(σ^{(n)}\) is the coefficient of the n field factors in \(j^{(n)}\). For a two-dimensional slab with vacuum along nonperiodic z, the following sheet quantities remove the dependence on the chosen vacuum height:
The units are \((m/V)^{n-1}\) for \(χ_{3D}\) and \(m^n/V^{n-1}\) for \(χ_{2D}\). Converge the basis set, the k mesh in the SCF calculation, and the k mesh used for the NLO susceptibility separately. Higher-order susceptibilities usually require large optical k meshes. [Edit on GitHub]
Keywords
Keyword descriptions
SECTION_PARAMETERS
Type: logical
Default: F
Lone keyword: T
Description: Activates the nonlinear optical susceptibility calculation. [Edit on GitHub]
ELEMENTS
Type: string[ ]
Default: -1
Repeatable: yes
Usage: ELEMENTS YXX YYY
Description: Susceptibility elements to compute and print. Use -1 alone for all elements (default), or a list of Cartesian index strings. Each string has ORDER+1 letters X, Y, or Z: the output direction followed by the input directions in FREQ_1 through FREQ_ORDER order. For example, YXX selects \(χ^{(2)}_{y;xx}\), and XYYYX selects \(χ^{(4)}_{x;yyyx}\). Uppercase and lowercase are accepted. Repeat the keyword to continue a long list. Duplicate entries are computed once. Only selected elements and their required intermediate responses are evaluated; unselected elements are omitted from the output. Do not combine -1 with explicit elements. [Edit on GitHub]
FREQ_1
Type: string[ ]
Usage: FREQ_1 RANGE 0.5 1.0 0.01
Description: Input photon energy in eV. VALUE specifies one energy; RANGE gives inclusive start, end, and positive step; SAME_AS and MINUS_OF reference an earlier input frequency. Independent ranges produce every combination. The sum of input energies must be nonzero. [Edit on GitHub]
FREQ_2
Type: string[ ]
Usage: FREQ_2 RANGE 0.5 1.0 0.01
Description: Input photon energy in eV. VALUE specifies one energy; RANGE gives inclusive start, end, and positive step; SAME_AS and MINUS_OF reference an earlier input frequency. Independent ranges produce every combination. The sum of input energies must be nonzero. [Edit on GitHub]
FREQ_3
Type: string[ ]
Usage: FREQ_3 RANGE 0.5 1.0 0.01
Description: Input photon energy in eV. VALUE specifies one energy; RANGE gives inclusive start, end, and positive step; SAME_AS and MINUS_OF reference an earlier input frequency. Independent ranges produce every combination. The sum of input energies must be nonzero. [Edit on GitHub]
FREQ_4
Type: string[ ]
Usage: FREQ_4 RANGE 0.5 1.0 0.01
Description: Input photon energy in eV. VALUE specifies one energy; RANGE gives inclusive start, end, and positive step; SAME_AS and MINUS_OF reference an earlier input frequency. Independent ranges produce every combination. The sum of input energies must be nonzero. [Edit on GitHub]
FREQ_5
Type: string[ ]
Usage: FREQ_5 RANGE 0.5 1.0 0.01
Description: Input photon energy in eV. VALUE specifies one energy; RANGE gives inclusive start, end, and positive step; SAME_AS and MINUS_OF reference an earlier input frequency. Independent ranges produce every combination. The sum of input energies must be nonzero. [Edit on GitHub]
K_MESH
Type: integer[3]
Usage: K_MESH 24 24 1
Description: Monkhorst-Pack-type uniform k mesh, shifted to include Γ, for the Brillouin-zone integration required for \(χ^{(n)}\). Fractional coordinates are k_i=j_i/N_i with j_i=0,…,N_i-1; no symmetry reduction is used. This mesh can differ from the required k-point SCF mesh. Higher-order nonlinear susceptibilities usually require denser k meshes. Every entry must be positive; use one in each nonperiodic direction. [Edit on GitHub]
ORDER
Type: integer
Default: 2
Usage: ORDER 2
Description: Order n of the electric-field response \(χ^{(n)}\); supported values are 2, 3, 4, and 5. Exactly FREQ_1 through FREQ_n must be set. [Edit on GitHub]
T1
Type: real
Default: [fs]
Usage: T1 [fs] 300
Description: Relaxation time for diagonal band occupations. Set together with T2 to use separate population and coherence times. Must satisfy T1 >= T2 > 0. Do not combine with an explicitly set T1_T2. [Edit on GitHub]
T1_T2
Type: real
Default: 3.00000000E+001 [fs]
Usage: T1_T2 [fs] 30
Description: Common relaxation time \(T_1=T_2=T\). Usually T₁ relaxes band occupations (the diagonal density-matrix elements \(ρ_{nn}\)) towards equilibrium; T₂ describes decoherence, damping off-diagonal elements \(ρ_{mn}\) to zero in the energy eigenbasis. Using one time gives \(-(ρ-ρ_{eq})/T\) and removes the basis-dependent distinction between populations and coherences inside degenerate subspaces. This provides a simple, numerically robust first approximation near band degeneracies and close bands. It does not establish that the physical lifetimes are equal. For relaxation-dependent singularities near the graphene Dirac point, see Cheng, Vermeulen and Sipe (2015). The broadening is \(Γ=ℏ/T_2\). Must be positive; the default is 30 fs. [Edit on GitHub]
References: Cheng2015
T1_T2_SMOOTHING_WIDTH
Type: real
Default: 2.50000000E-002 [eV]
Usage: T1_T2_SMOOTHING_WIDTH [eV] 0.025
Description: Positive energy width w of the smooth transition between population and coherence relaxation. The band-pair rate is 1/T1 + (1/T2 - 1/T1)[1 - exp(-((epsilon_m-epsilon_n)/w)^2)]. The default 0.025 eV is a model choice to converge for each system. [Edit on GitHub]
T2
Type: real
Default: [fs]
Usage: T2 [fs] 30
Description: Decoherence time for well-separated bands. Set together with T1. Near degenerate bands the relaxation rate changes smoothly to 1/T1. [Edit on GitHub]