The definition of polarization energy lowering in the original ALMO-EDA used the full AO space of each fragment as the variational degrees of freedom. This is based on the assumption that the AO basis functions are fragment-ascribable based on their atomic centers. However, this assumption becomes inappropriate when very large basis sets are used, especially those with diffuse functions (e.g. def2-QZVPPD). In such scenarios, basis functions on a given fragment tend to describe other fragments so that the “absolute localization" constraint becomes weaker and finally gets effectively removed. This is why the original ALMO-EDA scheme does not have a well-defined basis set limit for its polarization energy.
To overcome this problem, Horn and Head-Gordon
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J. Chem. Phys.
(2015),
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pp. 114111.
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proposed a new definition for
the POL term in the ALMO-EDA method based on fragment electrical response
functions (FERFs). FERFs on a given fragment are prepared by
solving CPSCF equations after its SCF solution is found:
(12.1) |
where is SCF orbital Hessian and is a component () of a multipole matrix with a certain order. The resulting fragment response matrices () are a set of matrices. Then, a singular value decomposition (SVD) is performed on :
(12.2) |
and the left vectors (not including the null vectors) will be used to construct a truncated virtual space, which is used to define the variational degrees of freedom for the SCF-MI problem:
(12.3) |
where denotes the original virtual orbitals of the given fragment.
The basic spirit of using FERFs is to obtain a subset of virtuals that is most pertinent to the electrical polarization of a given fragment, while the redundant variational degrees of freedom (which might be CT-like) are excluded. This scheme is shown to give a well-defined basis set limit for the polarization energy that relies on the SCF-MI calculation. The multipole orders dipole (D), quadrupole (Q), and octopole (O)m included on the RHS of Eq. (12.2), decide the span of FERFs on each fragment. Numerical experiments suggest that the inclusion of dipole- and quadrupole-type responses is able to long-range induced electrostatics correctly and also gives a well-defined basis set limit, which is thus recommended as the working basis of the SCF-MI problem. The full span of the polarization subspace of fragment is thus:
(12.4) |
Therefore, each occupied orbital will be paired with eight virtual orbitals (if the employed AO basis is large enough).
The polarization subspaces constructed as in Eq. (12.4) are
non-orthogonal between fragments. Therefore, it is named as the “nDQ" model
for polarization. There is another version of this method which enforces
interfragment orthogonality between the polarization subspaces and it is
correspondingly termed as “oDQ" (or with other multipole orders). The
preparation of orthogonal FERFs is more complicated (see
Ref.
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J. Chem. Phys.
(2015),
143,
pp. 114111.
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for the details) and usually gives less favorable
polarization energies. For most general cases, we recommend the use of the
“nDQ" model. Calculations using FERFs are performed using the generalized
SCF-MI procedure introduced in Section 12.6.2.
However, solving the CPSCF equation requires inverting the large orbital Hessian matrix of dimension , which is the time demanding step in solving for FERFs. For example, in Hartree-Fock theory, in the canonical representation, it has matrix elements
(12.5) |
Recently, Aldossary and coauthors proposed an uncoupled-FERFs (uFERFs) to improve the conventional FERFs. The idea of uFERFs is simply to adopt the solution for uncoupled CPSCF equation, where the orbital Hessian is approximated with
(12.6) |
With this approximation, the orbital Hessian is easily inverted, and the fragment response matrices are simply
(12.7) |
Benchmark calculations on the S22, S66 and Ionic43 datasets shows that the POL energy is hardly changed using uFERFs, while a general 5-10 times speedup for FERF creation and 20% average speedup for the whole EDA calculation is achieved. In addition, the importance of monopole uFERFs (response to scaled nuclear charges) for intermolecular interactions is recognized for strong ion-neutral interactions. Therefore, it is recommended to use MDQ-uFERFs for the calculation of POL energy in ALMO-EDA calculations.
CHILD_MP
CHILD_MP
Compute FERFs for fragments and use them as the basis for SCF-MI calculations.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE
Do not compute FERFs (use the full AO span of each fragment).
TRUE
Compute fragment FERFs.
RECOMMENDATION:
Use FERFs to compute polarization energy when large basis sets are used. In an “EDA2" calculation,
this $rem variable is set based on the given option automatically.
CHILD_MP_ORDERS
CHILD_MP_ORDERS
The multipole orders included in the prepared FERFs. The last digit specifies how many multipoles to compute,
and the digits in the front specify the multipole orders: 2: dipole (D); 3: quadrupole (Q); 4: octopole (O). Multipole
order 1 is reserved for monopole FERFs which can be used to separate the effect of orbital contraction.
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TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
21
D
232
DQ
2343
DQO
RECOMMENDATION:
Use 232 (DQ) when FERF is needed.
FRAG_CPSCF_MAXITER
FRAG_CPSCF_MAXITER
The maximum number of iterations executed by the conjugate-gradient solver before switching to the
MINRES algorithm. The maximum number of MINRES iterations is set to twice of the value of
FRAG_CPSCF_MAXITER. Note that this rem variable is also used for the CPSCF equations
involved in the adiabatic EDA (Sec. 12.7.3) and force decomposition analysis
(Sec. 12.7.4) calculations.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
User-defined
RECOMMENDATION:
Use the default
FRAG_CPSCF_CONV
FRAG_CPSCF_CONV
The convergence threshold for the CPSCF equation (using the RMS error of
the residual vector)
TYPE:
INTEGER
DEFAULT:
8
OPTIONS:
Convergence is reached when the RMS error is below
RECOMMENDATION:
Use the default
EDA_UFERF
EDA_UFERF
Using uncoupled-FERFs (uFERFs)
instead of FERFs
TYPE:
BOOLEAN
DEFAULT:
TRUE
OPTIONS:
TRUE
Use uFERFs
FALSE
Use FERFs
RECOMMENDATION:
Use the default uFERFs, use FALSE when FERFs are desired.
Example 12.13 Generalized SCF-MI calculation for the water dimer using nDQ FERFs.
$molecule 0 1 -- 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 -- 0 1 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561 $end $rem METHOD wb97x-v GEN_SCFMAN true BASIS 6-31+G(d) GEN_SCFMAN true SCF_ALGORITHM diis SCF_CONVERGENCE 8 THRESH 14 SCF_FINAL_PRINT 1 EDA_UFERF false FRGM_METHOD stoll SCFMI_MODE 1 !nonortho gen scfmi CHILD_MP true CHILD_MP_ORDERS 232 !DQ FD_MAT_VEC_PROD false INTEGRAL_SYMMETRY false POINT_GROUP_SYMMETRY false $end