Manual:PROPERTIES:LINEAR RESPONSE
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Linear Response module written by T. Saue and H. J. Aa. Jensen [1].
The general form of a linear response function in the random phase approximation (RPA) is [1] [2]
![\langle\langle A ; B \rangle\rangle_{\omega} = -\mathbf{E}_A^{[1]\dagger}
\bigl(
E^{[2]} - \omega S^{[2]}
\bigr)^{-1}
\mathbf{E}_B^{[1]}](/dirac/images/math/3/3/a/33ad8ef666cb0f7d49ecd15b16261f82.png)
where
![\mathbf{E}_A^{[1]}](/dirac/images/math/6/3/a/63aec7d1787fddef0b15ae2969fc35f5.png)
and
![\mathbf{E}_B^{[1]}](/dirac/images/math/2/1/8/2182a1c97636f06f80b2642613fe3de6.png)
are property gradients corresponding to properties A and B,
E[2]
is the molecular Hessian,
S[2]
is the metric, and
ω
is a frequency.
The dimension of
E[2]
does in general not allow an explicit inversion of the resolvent
![\bigl(
E^{[2]} - \omega S^{[2]}
\bigr)^{-1}](/dirac/images/math/6/c/b/6cbb112cbbbb544ae13bedb409423498.png)
, so one resorts to iterative techniques.
We write:
![\bigl(
E^{[2]} - \omega S^{[2]}
\bigr)^{-1}
\mathbf{E}_B^{[1]}
= \mathbf{X}](/dirac/images/math/e/5/8/e589ae342edd882b5ae789f3aec83ddc.png)
By a slight rearrangement we arrive at the linear response equations:
![\bigl(
E^{[2]} - \omega S^{[2]}
\bigr)
\mathbf{X}
= \mathbf{E}_B^{[1]}](/dirac/images/math/2/1/b/21b88e00d350c78835867c666a7b7015.png)
The RPA equation is solved by expanding 
in a set of trial vectors:
![\mathbf{X} = \sum_{i=1} \mathbf{b}_i a_i;
Y = [\mathbf{b}_1 \mathbf{b}_2 \cdots \mathbf{b}_n]](/dirac/images/math/0/7/b/07bb04070d9439e9ff637776dcd5b50b.png)
and solving the reduced equation:
![\bigl(
\tilde{E}^{[2]} - \omega \tilde{S}^{[2]}
\bigr)
\mathbf{a}
= \tilde{\mathbf{E}}_B^{[1]}](/dirac/images/math/e/1/1/e11f9a918dde36f49e11ade054b1fb3e.png)
in which appear:
![\tilde{E}^{[2]} = Y^\dagger E^{[2]} Y;
\tilde{S}^{[2]} = Y^\dagger S^{[2]} Y;
\tilde{\mathbf{E}}_B^{[1]} = Y^\dagger \mathbf{E}_B^{[1]}](/dirac/images/math/9/b/2/9b2aa6df9f5934136d760e64b1c8fe7d.png)
General control statements
Print level.
Default:
.PRINT 0
Definition of the linear response function
.A OPERATOR
Specification of the A operator (see the specification of one-electron operators for details).
.B OPERATOR
Specification of the B operator (see the specification of one-electron operators for details).
.OPERATORS
Specification of both the A and B operators (see the specification of one-electron operators for details).
.TRIAB
Enforce triangularity of response function.
Only one function is calculated.
Default: Deactivated.
.B FREQ
Specify frequencies of operator B.
Example: 3 different frequencies.
.B FREQ 3 0.001 0.002 0.01
Default: Static case.
.B FREQ 1 0.0
.IMAGIN
Employ imaginary frequencies.
.ALLCMB
Form all possible
even if imaginary.
Default: Deactivated.
.UNCOUP
Uncoupled calculation.
Control variational parameters
.OCCUP
For each fermion ircop give an orbital string of inactive orbitals to include in the linear response calculation.
.VIRTUA
For each fermion ircop give an orbital string of virtual orbitals to include in the linear response calculation.
.SKIPEE
Exclude all rotations between occupied positive-energy and virtual positive-energy orbitals.
.SKIPEP
Exclude all rotations between occupied positive-energy and virtual negative-energy orbitals.
Control reduced equations
.MAXITR
Maximum number of iterations.
Default:
.MAXITR 30
.MAXRED
Maximum dimension of matrix in reduced system.
Default:
.MAXRED 200
.THRESH
Threshold for convergence of reduced system.
Default:
.THRESH 1.0D-5
Control integral contributions
The user is encouraged to experiment with these options since they may have an important effect on run time.
.INTFLG
Specify what two-electron integrals to include (see .INTFLG under **HAMILTONIAN).
Default: .INTFLG from **HAMILTONIAN.
.CNVINT
Set threshold for convergence before adding SL and SS integrals to SCF-iterations.
2 (real) Arguments:
.CNVINT CNVXQR(1) CNVXQR(2)
Default: Very large numbers.
.ITRINT
Set the number of iterations before adding SL and SS integrals to SCF-iterations.
Default:
.ITRINT 1 1
Control trial vectors
.REAXVC
Read solution vectors from file XVCFIL
.REAXVC
XVCFIL
Default: No restart on solution vectors. The file has to have six characters. Make sure there is no blank character in front of the file name.
For a restart on solution vectors it is useful to set
.REAXVC
XVCFIL
.ITRINT
0 0
otherwise LS-integrals (and SS-integrals) are switched on later and one may first iterate away and then back to a possibly converged response vector.
Often you have a converged SCF wave function along with a response vector. In this case make sure that
**DIRAC #.WAVE FUNCTION
is commented out. Make then also sure that you use the DFCOEF file which has been obtained in the same calculation as the response vector file. Otherwise you may observe more response solver iterations than necessary.
.XLRNRM
Normalize trial vectors. Using normalized trial vectors will reduce efficiency of screening.[please clarify]
Default: Use un-normalized vectors.
Advanced/debug flags
.E2CHEK
Generate a complete set of trial vector which implicitly allows the explicit construction of the electronic Hessian. Only to be used for small systems !
.ONLYSF
Only call FMOLI in sigmavector routine: only generate one-index transformed Fock matrix [1].
.ONLYSG
Only call FMOLI in sigmavector routine: 2-electron Fock matrices using one-index transformed densities [1].
.STERNHEIM
Set diagonal elements of orbital part of Hessian equal to − 2mc2 for rotations between occupied positive-energy and virtual negative-energy orbitals.
Default: Deactivated.
.STERNC
(Sternheim complement) allows to separate basis set incompleteness from the replacement of an inner sum over negative-energy orbitals only by the full sum. In order to benefit from this functionality (only for specialists !), you should run with print level 2 under properties.
Then you can do a sequence of calculations: 1) .SKIPEP 2) .STERNH 3) .STERNC The diamagnetic contribution of 1) is the non-relativistic expectation value, whereas 2) is the Sternheim approximation, that is replacing orbital energy differences with − 2mc2. With no basis set incompleteness the sum of the diamagnetic contribution 2) and the paramagnetic contribution 3) should equal the diamagnetic contribution of 1).
Default: Deactivated.
.COMPRESSION
Reduce number of orbital variation parameters by checking corresponding elements of gradient vector against a threshold. This may reduce memory.
Default: No compression.
.COMPRESSION 0.0
.NOPREC
No preconditioning of initial trial vectors.
Default: Preconditioning of trial vectors.
.RESFAC
New trial vector will be generated only for variational parameter classes whose residual has a norm that is larger than a fraction 1/RESFAC of the maximum norm.
Default:
.RESFAC
1000.0
References
- ↑ 1.0 1.1 1.2 1.3 T. Saue and H. J. Aa. Jensen, Linear response at the 4-component relativistic level: Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers, J. Chem. Phys. 118, 522 (2003) electronic version.
- ↑ T. Saue, Post Dirac-Hartree-Fock methods - Properties, in Relativistic Electronic Structure Theory - Part 1: Fundamentals, edited by P. Schwerdtfeger, Elsevier, Amsterdam, 2002.
