7.16. CASSCF Linear Response¶
Similar to the SCF linear response (see CP-SCF Options), second-derivative properties can be calculated at the CASSCF level by solving the coupled perturbed (CP-)CASSCF equations for the linear response of the wavefunction parameters to a perturbation. These linear response equations are expressed as
where \(\boldsymbol\lambda\) are the CASSCF wavefunction parameters and \(\mathbf R\) are the perturbations for which the response equations are being solved [378, 861]. The property gradient on the right-hand side (RHS) is a known perturbation-dependent quantity, but the left-hand side (LHS) depends on the solution of the response of the wavefunction parameters to \(\mathbf R\). Therefore, the response to the perturbation must be solved for iteratively, which is done using the trial vectors \(\mathbf X\). This leads to the LHS being computed as a sigma-vector, which is reassembled at every iteration. The linear response equation for perturbation \(R_i\) can then be written as
where \(\mathbf H\) is the perturbation-independent Hessian matrix and \(\mathbf G\) is the perturbation-specific RHS matrix. Once the linear equation converges, the solution vector of the response parameters can then be contracted with the electron- and spin-density matrices to yield the AO response density matrices, \(\left[\mathbf P^{\alpha \pm \beta}\right]^{(\mathbf R)}\). The response densities can then be contracted with appropriate AO 1-electron property integrals to compute the second-order contributions to second-derivative properties.
In ORCA these equations are solved by the orca_casscfresp program and the underlying solver is BHP22, a Davidson-type linear equation solver.
The RHS is built from the property integrals calculated in the orca_propint program.
After the solution converges and the response densities are made and stored, the orca_prop program is called, wherein the appropriate densities and response densities are contracted with the necessary property integrals (also from orca_propint).
This use of densities keeps the response property calculations in orca_prop generally applicable to all methods as the densities house the method-specific information.
7.16.1. Input Block¶
The input block %casresp is available for requesting the following options (given below with their default values):
%casresp
TolR 1.e-5 # Convergence tolerance for the residual norm
# Note: TolR not affected by global keywords (TightSCF, etc.)
MaxIter 64 # Maximum number of iterations
MaxRed 24 # Maximum size of the reduced space per RHS
PrintRHSVec false # Print significant contributions to the RHS vector
PrintRspVec false # Print significant contributions to the response vector
PrintWF CFG # print CI part in (CFG (default), CSF, or DET basis)
TolPrintVec 1.e-2 # Threshold to print a contribution to the RHS/response vector
# Minimum value of the SQUARE of a vector element to print
PreCondType diag # (default) Use diagonal preconditioner (requires auxiliary basis!)
none # Do not use a preconditioner
PreCondMaxRed 400 # Hessian precondition subset size
end
Note that the solver-related options in the %elprop and %eprnmr blocks do not affect the solution of the CP-CASSCF equations.
7.16.2. Technical Notes¶
It should be noted that CASSCF linear response uses the optimized CASSCF wavefunction as a starting point.
Thus, a %casscf block with the appropriate inputs (see Complete Active Space Self-Consistent Field Method) must be provided in the input.
State-specific (SS-)CASSCF response is run on SS (NRoots 1) CASSCF wavefunctions.
If the CASSCF wavefunction is state-averaged (SA), the response is run over the averaged manifold of states and not on a specific root.
In this case, one should analyze the output carefully.
It is wise to only average states which are (nearly-)degenerate.
By default, a CASSCF calculation will be run from scratch before running the CASSCF Response.
Alternatively, orbitals from a previously converged CASSCF calculation may be used with !MOREAD NoIter and supplying the appropriate .gbw file via %moinp.
In this case, be sure that the input in the %casscf block matches that from the same input block of the previously converged calculation and be sure to check the orbitals well!
The appropriate property flags in the %elprop and %eprnmr blocks must be set to calculate the properties that are wanted (see Electric Properties, EPR and NMR properties, and Calculation of Properties).
While all first-derivative properties work with CASSCF, not all second-derivative properties are currently available with CASSCF.
The static properties currently available (sorted by perturbation taken for the response) include:
Electric (\(\mathbf E\)) Field
Dipole/dipole Polarizability
Dipole/quadrupole Polarizability
Quadrupole (\(\mathbf Q\)) Field
Quadrupole/Quadrupole Polarizability
Magnetic (\(\mathbf B\)) Field (without GIAOs)
EPR g-Tensor
NMR Chemical Shieldings
Velocity (\(\mathbf v\))
Velocity Polarizability
The magnetic properties are currently implemented without gauge-including atomic orbitals (GIAOs).
Thus, an appropriate gauge-origin must be provided in the %eprnmr block!
For the EPR g-tensor, using a large basis set with a chemically relevant gauge origin is recommended.
However, one should in general be wary of NMR chemical shieldings without GIAOs.
GIAOs for CASSCF linear response are coming soon to ORCA!
7.16.3. Notes on Printing¶
The information on the types of property integrals computed can be found in the ORCA PROPERTY INTEGRAL CALCULATIONS section of the output file.
The orca_casscfresp output begins with the header ORCA CASSCF RESPONSE CALCULATION.
After information about the types and number of perturbations, the calculation splits into the major types of perturbations: real and imaginary.
All response equations of the same type can be solved simultaneously.
Each of these types has its own section in the output file. These sections begin with information on the orbital ranges and the CI space, then go into the iterative solution of the equations. The printout here gives an overview of the iteration number, the residual norm of each response equation, and if that response equation has met the convergence criteria. The following is an example of this output (with iterations 3–7 removed for simplicity):
--------------------------------------------------------------------------------------------
LINSOL Davidson-type linear equation solver
Iter. ||Error||_2 Conv. (TolR = 1.000e-08)
--------------------------------------------------------------------------------------------
0 (rhs 0) 1.645417e-01 No
(rhs 1) 1.826041e-01 No
(rhs 2) 2.047543e-01 No
1 (rhs 0) 7.069575e-02 No
(rhs 1) 3.363295e-02 No
(rhs 2) 7.229274e-02 No
2 (rhs 0) 9.183136e-03 No
(rhs 1) 3.922435e-03 No
(rhs 2) 7.152764e-03 No
(...)
8 (rhs 0) 4.070240e-09 Yes
(rhs 1) 1.186846e-09 Yes
(rhs 2) 4.407666e-09 Yes
Before going on to the necessary property modules, the significant contributions to the RHS and/or response vectors will be outputted if they were requested (via PrintRHSVec and PrintRspVec).
Here, outputs such as the following can be seen.
---------------------------------
CASSCF Response Vector Analysis
---------------------------------
square of coefficients of particle-hole and CAS-CI excitations are printed if larger than 1.0e-02
IPERT: 0
(-) I 9 ( 9) -> V 0 ( 17) : 0.07167 (c= -0.26772137)
(...)
(-) A 1 ( 14) -> V 13 ( 30) : 0.07323 (c= 0.27061481)
(...)
(-) I 4 ( 4) -> A 0 ( 13) : 0.02915 (c= 0.17074032)
(...)
(-) CI 1.65592 [ 7]: 1201
(...)
The majority of the output has been removed for the sake of simplicity.
Here, the significant contributions to the response vector are printed if the square of the vector element is larger than 1.0e-02 (the default value for TolPrintVec).
Each line begins with either a (+) or (-), which denote the Hermiticity of the excitation operator.
With static CASSCF linear response, (+) always denotes an imaginary perturbation and (-) a real perturbation.
For the first vector (i.e. IPERT: 0), four contributions have been listed here.
The first three are from particle-hole excitations going from the left size of the arrow to the right.
On each side of the arrow has three things: a letter designating if it is an inactive (I), active (A), or virtual (V) orbital; a number designating the index within that orbital subblock, and a number in parentheses designating the overall orbital number.
Further to the right, the coefficient (i.e. the vector element) is given in parentheses and the square of that value is given to the left of it.
This is how the relative contributions of each excitation can be analyzed.
The final contribution shown is from a CI excitation. Its weight is listed, followed by the index in brackets and the corresponding configuration of the active electrons among the active orbitals.
7.16.4. Troubleshooting¶
If the %casresp block is specified and the calculation does not run through the CASSCF Response section, then the properties requested are not second-derivatives and therefore do not require linear response equations to be solved.
If, however, the job aborts, watch for these possibilities:
At least one of the second-derivative properties requested is not available at the CASSCF level (see list above of those currently implemented)
MaxItermay need to be increased orTolRdecreasedA magnetic property is requested without setting the gauge-origin in the
%eprnmrblockAn appropriate auxiliary basis (or the
!AutoAuxkeyword) may be required, especially for:The diagonal preconditioner (which is on by default)
TrafoStep RIin the%casscfblockRIJCOSX, RIJONX
RIJK was specified (this is NOT supported)
7.16.5. Minimal Input File¶
The following is an overview of the blocks required for a property calculation with
! def2-TZVP AutoAux # (or other appropriate basis & auxiliary)
%casscf
...
end
%casresp # only required to change solver, printing, or preconditioner options
...
end
%elprop # only required if electric property requested
...
end
%eprnmr # only required if magnetic property requested
...
ORI ...
end
* xyzfile ...