The Effective Fragment Potential Method

The basic idea behind the effective fragment potential (EFP) method is to replace the chemically inert part of a system by EFPs, while performing a regular ab initio calculation on the chemically active part. Here "inert" means that no covalent bond breaking process occurs. This "spectator region" consists of one or more "fragments", which interact with the ab initio "active region" through non-bonded interactions, and so of course these EFP interactions affect the ab initio wavefunction. A simple example of an active region might be a solute molecule, with a surrounding spectator region of solvent molecules represented by fragments. Each discrete solvent molecule is represented by a single fragment potential, in marked contrast to continuum models for solvation.

The quantum mechanical part of the system is entered in the $DATA group, along with an appropriate basis. The EFPs defining the fragments are input by means of a $EFRAG group, one or more $FRAGNAME groups describing each fragment's EFP, and a $FRGRPL group. These groups define non-bonded interactions between the ab initio system and the fragments, and between the fragments. The former interactions enter via one-electron operators in the ab initio Hamiltonian, while the latter interactions are treated by analytic functions. The only electrons explicitly treated (e.g. with basis functions used to expand occupied orbitals) are those in the active region, so there are no new two electron terms. Thus the use of EFPs leads to significant time savings compared to full ab initio calculations on the same system.

At ISU, the EFPs are currently used to model RHF/DZP water molecules in order to study aqueous solvation effects, for example references 1,2,3. Our co-workers at NIST have also used EFPs to model parts of enzymes, see reference 4.

Terms in an EFP

The non-bonded interactions currently implemented are:

  1. Coulomb interaction. The charge distribution of the fragments is represented by an arbitrary number of charges, dipoles, quadrupoles, and octupoles, which interact with the ab initio hamiltonian as well as with multipoles on other fragments. It is possible to input a screening term that accounts for the charge penetration. Typically the multipole expansion points are located on atomic nuclei and at bond midpoints.
  2. Dipole polarizability. An arbitrary number of dipole polarizability tensors can be used to calculate the induced dipole on a fragment due to the electric field of the ab initio system as well as all the other fragments. These induced dipoles interact with the ab initio system as well as the other EFPs, in turn changing their electric fields. All induced dipoles are therefore iterated to self-consistency. Typically the polarizability tensors are located at the centroid of charge of each localized orbital of a fragment.
  3. Repulsive potential. Two different forms for the repulsive potentials are used: one for ab initio-EFP repulsion and one for EFP-EFP repulsion. The form of the potentials is empirical, and consists of distributed gaussian or exponential functions, respectively. The primary contribution to the repulsion is the quantum mechanical exchange repulsion, but the fitting technique used to develop this term also includes the effects of charge transfer. Typically these fitted potentials are located on atomic nuclei within the fragment.

Constructing an EFP Using GAMESS

RUNTYP=MOROKUMA assists in the decomposition of inter- molecular interaction energies into electrostatic, polarization, charge transfer, and exchange repulsion contributions. This is very useful in developing EFPs since potential problems can be attributed to a particular term by comparison to these energy components for a particular system.

A molecular multipole expansion can be obtained using $ELMOM. A distributed multipole expansion can be obtained by either a Mulliken-like partitioning of the density (using $STONE) or by using localized molecular orbitals ($LOCAL: DIPDCM and QADDCM). The molecular dipole polarizability tensor can be obtained during a Hessian run ($CPHF), and a distributed LMO polarizability expression is also available ($LOCAL: POLDMC).

The repulsive potential is derived by fitting the difference between ab initio computed intermolecular interaction energies, and the form used for Coulomb and polarizability interactions. This difference is obtained at a large number of different interaction geometries, and is then fitted. Thus, the repulsive term is implicitly a function of the choices made in representing the Coulomb and polarizability terms. Note that GAMESS currently does not provide a way to obtain these repulsive potential, or the charge interpenetration screening parameters.

Since you cannot develop all terms necessary to define a new EFP's $FRAGNAME group using GAMESS, in practice you will be limited to using the internally stored H2OEF2 potential mentioned below.

Current Limitations

  1. The energy and energy gradient are programmed, which permits RUNTYP=ENERGY, GRADIENT, and numerical HESSIAN. The necessary modifications to use the EFP gradients while moving on the potential surface are programmed for RUNTYP=OPTIMIZE, SADPOINT, and IRC (see reference 3), but the other gradient based potential surface explorations such as DRC are not yet available. Finally, RUNTYP=PROP is also permissible.
  2. The ab initio system must be treated with RHF, ROHF, UHF, or the open shell SCF wavefunctions permitted by the GVB code. The correlated methods in GAMESS (MP2, GVB-PP, CI, MCSCF) should not be used, since the available H2OEF2 potential was derived at the RHF level, and therefore does not contain dispersion terms. A correlated computation on the ab initio system without these terms in the EFP will probably lead to unphysical results.
  3. EFPs can move relative to the ab initio system and relative to each other, but the internal structure of an EFP is frozen.
  4. The boundary between the ab initio system and the EFPs must not be placed across a chemical bond.
  5. Calculations must be done in C1 symmetry at present.
  6. If you need IR intensities, you have to use NVIB=2. The potential surface is usually very soft for EFP motions, and double differenced Hessians should usually be obtained.

Practical hints for using EFPs

At the present time, we have only one EFP suitable for general use. This EFP models water, and its numerical parameters are internally stored, using the fragment name H2OEF2. These numerical parameters are improved values over the H2OEF1 set which were presented and used in reference 2, and they also include the improved EFP-EFP repulsive term defined in reference 3. The H2OEF2 water EFP was derived from RHF/DH(d,p) computations on the water dimer system. When you use it, therefore, the ab initio part of your system should be treated at the SCF level, using a basis set of the same quality (ideally DH(d,p), but probably other DZP sets such as 6-31G(d,p) will give good results as well). Use of better basis sets than DZP with this water EFP has not been tested.

As noted, effective fragments have frozen internal geometries, and therefore only translate and rotate with respect to the ab initio region. An EFP's frozen coordinates are positioned to the desired location(s) in $EFRAG as follows:

  1. the corresponding points are found in $FRAGNAME.
  2. Point -1- in $EFRAG and its FRAGNAME equivalent are made to coincide.
  3. The vector connecting -1- and -2- is aligned with the corresponding vector connecting FRAGNAME points.
  4. The plane defined by -1-, -2-, and -3- is made to coincide with the corresponding FRAGNAME plane.
Therefore the 3 points in $EFRAG define only the relative position of the EFP, and not its internal structure. So, if the "internal structure" given by points in $EFRAG differs from the true values in $FRAGNAME, then the order in which the points are given in $EFRAG can affect the positioning of the fragment. It may be easier to input water EFPs if you use the Z-matrix style to define them, because then you can ensure you use the actual frozen geometry in your $EFRAG. Note that the H2OEF2 EFP uses the frozen geometry r(OH)=0.9438636, a(HOH)=106.70327, and the names of its 3 fragment points are ZO1, ZH2, ZH3.

The translations and rotations of EFPs with respect to the ab initio system and one another are automatically quite soft degrees of freedom. After all, the EFP model is meant to handle weak interactions! Therefore the satisfactory location of structures on these flat surfaces will require use of a tight convergence on the gradient: OPTTOL=0.00001 in the $STATPT group.

References

The first of these is more descriptive, and the second has a very detailed derivation of the method. The latest EFP developments are discussed in the 3rd paper.

  1. "Effective fragment method for modeling intermolecular hydrogen bonding effects on quantum mechanical calculations" J.H.Jensen, P.N.Day, M.S.Gordon, H.Basch, D.Cohen, D.R.Garmer, M.Krauss, W.J.Stevens in "Modeling the Hydrogen Bond" (D.A. Smith, ed.) ACS Symposium Series 569, 1994, pp 139-151.
  2. "An effective fragment method for modeling solvent effects in quantum mechanical calculations". P.N.Day, J.H.Jensen, M.S.Gordon, S.P.Webb, W.J.Stevens, M.Krauss, D.Garmer, H.Basch, D.Cohen J.Chem.Phys. 105, 1968-1986(1996).
  3. "The effective fragment model for solvation: internal rotation in formamide" W.Chen and M.S.Gordon, J.Chem.Phys., submitted.
  4. "Transphosphorylation catalyzed by ribonuclease A: Computational study using ab initio EFPs" B.D.Wladkowski, M. Krauss, W.J.Stevens, J.Am.Chem.Soc. 117, 10537-10545(1995).

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