QM/FQ: Quantum Mechanics/Fluctuating Charges

The Quantum Mechanics/Fluctuating Charges (QM/FQ) method is a multiscale model designed to describe the properties of a chemical system perturbed by the presence of its molecular environment, where the latter described using a polarizable classical force-field, see for example Refs. [1] [2] [3]. The QM/FQ model has been applied to the calculation of spectroscopic properties of molecules in aqueous solution. Each atom of the environment (e.g. a solvation shell) is endowed with a fluctuating charge (FQ) which can vary in response to the electrostatic potential generated by the solute. The FQ charges can be constrained such that each molecule within the environment remains electrically neutral. The energy functional of the FQ system can be written as:

\[F(q,\lambda) = \sum_{\alpha, i} q_{\alpha, i} \chi_{\alpha, i} + 1/2\sum_{\alpha, i}\sum_{\beta, j} q_{\alpha, i} J_{\alpha i, \beta j} q_{\beta, j} + \sum_{\alpha}\lambda_\alpha (\sum_i q_{\alpha, i} - Q_\alpha)\]

where \(q_{\alpha, i}\) is the i-th FQ charge belonging to the \(\alpha\)-th molecule, \(\chi_{\alpha,i}\) is the electronegativity of each atom, \(\lambda_\alpha\) is the Lagrange multiplier associated with the \(\alpha\)-th molecule, whose purpose is to fix the total charge to be \(Q_\alpha\). The matrix \(J_{\alpha i, \beta j}\) describes the interaction kernel between the FQs: while the diagonal elements are related to the self-interaction through the chemical hardnesses \(eta\), the off-diagonal elements may be specified through different formulations. The FQ charges can be obtained by minimizing the energy functional with respect to the charges and the Lagrange multipliers, which leads to the following set of linear equations:

\[Dq_\lambda = -C_Q\]

where \(C_Q\) is the vector of electronegativities and charge constraints, \(q_\lambda\) contains the FQ charges and the Lagrange multipliers, while D is the interaction matrix that contains the kernel J and the Lagrangian blocks.

The coupling of the FQ model with a QM Hamiltonian can be done by introducing the QM/FQ interaction operator as

\[H_\text{QM/FQ} = \sum_i^{N} q_i V[\rho](r_i),\]

where \(N\) is the number of FQ charges, \(q_i\) is the i-th FQ charge located at position \(r_i\) and \(V[\rho](r_i)\) is the electric potential generated by the QM system on the same point. The introduction of the QM/FQ interaction leads to a modified set of linear equations for the FQ charges, i.e.

\[Dq_\lambda = -C_Q-V\]

where V is the electric potential generated by the QM electrons and nuclei at the position of each FQ charge. The FQ charges are thus determined self-consistently along with the ground-state density. Since the charges depend on the QM density, explicit terms also appear within response equations that are solved to simulate spectroscopic and excited-state properties of the QM system.

Starting from AMS2021.103 a screening is included for the interaction between MM atoms and the QM density, to avoid unstable results in case numerical integration points are accidentally close to MM atoms. The screened \(1/r_{ij}\) has the form:

\[\begin{split}\text{ERF: =} & \frac{erf(ar_{ij})}{r_{ij}} \\ \text{EXP: =} & \frac{(1-exp(-ar_{ij}))^2}{r_{ij}} \\ \text{GAUS: =} & \frac{erf(r_{ij}/a)}{r_{ij}}\end{split}\]

where \(a\) = QMSCREENFACTOR.

Input options

FQQM
   DEBUG Yes/No
   QMSCREEN [ERF | EXP | GAUS | NONE]
   QMSCREENFACTOR float
   SCREEN [COUL | OHNO | GAUS]
   TOTALCHARGE float
End
FQQM
Type:Block
Description:Block input key for QM/FQ.
DEBUG
Type:Bool
Default value:No
Description:The DEBUG subkey will print additional information from the FQ subroutines.
QMSCREEN
Type:Multiple Choice
Default value:GAUS
Options:[ERF, EXP, GAUS, NONE]
Description:Expert option. QMSCREEN can be used to choose the functional form of the charge-charge interaction kernel between MM atoms and the QM density. The screening types available are ERF (error function), EXP (exponential), GAUS (Gaussian), or NONE. The default is GAUS.
QMSCREENFACTOR
Type:Float
Default value:0.2
Description:Expert option. Sets the QM/MM interaction kernel screening length. Recommended is to use the default value 0.2 with the GAUS QM/MM screening function.
SCREEN
Type:Multiple Choice
Default value:OHNO
Options:[COUL, OHNO, GAUS]
GUI name:Screen
Description:Expert option. SCREEN can be used to choose the functional form of the charge-charge interaction kernel between MM atoms. Recommended is to use the default OHNO. The COUL screening is the standard Coulomb interaction 1/r. The OHNO choice introduce the Ohno functional (see [K. Ohno, Theoret. Chim. Acta 2, 219 (1964)]), which depends on a parameter n that is set equal to 2. Finally, the GAUS screening models each FQ charge by means of a spherical Gaussian-type distribution, and the interaction kernel is obtained accordingly.
TOTALCHARGE
Type:Float
Default value:0.0
GUI name:Total charge on each FQ molecule
Description:The TOTALCHARGE subkey rules the charge constraint on each FQ molecule. 
FQPAR
  Element
    CHI value
    ETA value
  SUBEND
  GROUP groupname
    natoms
     elem x.xxx y.yyy z.zzz
     elem x.xxx y.yyy z.zzz
     elem x.xxx y.yyy z.zzz
     ...
  SUBEND
END
Element

Within the FQPAR block, you will need a sub-block that defines the parameters for each element that is in your FQ system. You will need to replace ‘Element’ with the element you are assigning parameters to, as in:

Ag
  ...
SUBEND

if you are assigning parameters to Ag. Note that the first letter MUST be capitalized and the second MUST be lowercase.

CHI value
CHI specifies the atomic electronegativity (in a.u.)
ETA value
ETA specifies the chemical hardness (in a.u.)

In case of water recommended is to use the optimized FQ parameters CHI and ETA for O and H in water, the so called Giovannini parameters [5]

O
  CHI 0.189194
  ETA 0.523700
SUBEND
H
  CHI 0.012767
  ETA 0.537512
SUBEND

Alternatively, in case of water as solvent, one could use the so called Rick parameters [6]

O
  CHI 0.116859
  ETA 0.584852
SUBEND
H
  CHI 0.000001
  ETA 0.625010
SUBEND
GROUP groupname

The GROUP sub-block is where the FQ atom coordinates are given. A (unique) groupname is required (maximum 10 characters).

Example for a water molecule:

GROUP water1
3
O  0.00000  0.00000  0.59372
H  0.00000  0.76544 -0.00836
H  0.00000 -0.76544 -0.00836
SUBEND

The first line gives the number of atoms to follow. Every line after that contains the element in the first column (first letter MUST be capitalized, second MUST be lowercase), then the x-component, then the y-component, then the z-component. The parameters for each element should have been defined in the ‘Element’ sub-block at the beginning of the FQPAR section.

References

[1]T. Giovannini, F. Egidi, C. Cappelli, Molecular spectroscopy of aqueous solutions: a theoretical perspective, Chemical Society Reviews, 49, 5664 (2020)
[2]C. Cappelli, Integrated QM/polarizable MM/continuum approaches to model chiroptical properties of strongly interacting solute–solvent systems, International Journal of Quantum Chemistry, 116, 1532 (2016)
[3]T. Giovannini, F. Egidi, C. Cappelli, Theory and algorithms for chiroptical properties and spectroscopies of aqueous systems, Physical Chemical Chemical Physics, 22, 22864 (2020)
[4]K. Ohno, Some remarks on the Pariser-Parr-Pople method, Theoretica Chimica Acta, 2, 219 (1964)
[5]T. Giovannini, P. Lafiosca, B. Chandramouli, V. Barone, C. Cappelli, Effective yet reliable computation of hyperfine coupling constants in solution by a QM/MM approach: Interplay between electrostatics and non-electrostatic effects, Journal of Chemical Physics 150 (2019) 124102
[6]S.W. Rick, S.J. Stuart, B.J. Berne, Dynamical fluctuating charge force fields: Application to liquid water, Journal of Chemical Physics 101 (1994) 6141