17.5 Polarizable embedding calculations

Polarizable embedding (PE) calculations are a based on a hybrid model of quantum mechanics and molecular mechanics (QM/MM) in which the classical region is represented by an electrostatic potential with up to octupole moments and induced point dipole moments. The main improvement over the more common QM/MM approaches without polarizable MM sites can be found for the description of electronic excitations but also for any other process which causes a significant change in the QM density and which is accompanied by a fast response of the environment.

In TURBOMOLE, only ground state energies computed with the dscf, ridft, and ricc2 module and electronic excitation properties based on RI-CC2 are implemented. The general theory is presented in ref.  [208] and [209], the PERI-CC2 model and the TURBOMOLE implementation is described in ref.  [210].

17.5.1 Theory

In the following, only the most important ideas are presented and discussed with a focus on the PERI-CC2 model. The essential concept is the introduction of an environment coupling operator Ĝ(DCC)

Gˆ(DCC ) =   ˆGes + ˆGpol(DCC )                  (17.13)

with the electrostatic contribution

         M  K
ˆGes  =  ∑   ∑  ∑  Θ(k) Q(k)ˆEpq                 (17.14)
        m=1 k=0pq  m,pq m

and the polarization contribution

ˆ pol  CC      ∑U  ∑   (1)  ind   CC ˆ
G   (D   )  =         Θu,pqμu (D   )Epq.             (17.15)
              u=1 pq

Here, Θm,pq(k) are multipole interaction integrals of order k and μuind are the induced dipoles which can be obtained from the electric field Fu and the polarizability αu at a site u:

μind= F  α
  u     u u

Because the induced dipoles depend on the electron density and vice versa, their computations enter the self-consistent part of the HF cycle. Introducing Ĝ(DCC) into standard equations for the HF reference state and the CC2 equations leads to a general PE-CC2 formulation. To maintain efficiency, a further approximation has been introduced which makes the operator only dependent on a CCS-like density term. These general ideas define the PERI-CC2 model and allow to formulate the corresponding Lagrangian expression

                              ˆ  ˆ   ˆ   1 ˆ2
LPERI-CC2(t,t)  =  EPE-HF + ⟨HF |W  (T1 + T2 + 2T1 )|HF⟩ +
                  ∑  t  ⟨μ | ˜W + [ˆFPE,Tˆ]+ [ ˜W ,T ˆ]|HF⟩+
                   μ1 μ1 1           1        2
                  ∑              PE
                     tμ2⟨μ2| ˜W + [ˆF ,Tˆ2]|HF ⟩
                  - 1∑  F elec(D Δ′)RuvF elec(D Δ′)             (17.17)
                    2 uv  u           v

from which all PERI-CC2 equations including the linear response terms may be derived. Note that the dependency on the density couples the CC amplitude and multiplier equations for the ground state solution vector.

This coupling is avoided by the simplified polarizable embedding method (sPE) described in ref.  [211].

LsPERI- CC2(t,t) =EPE -HF
                          PE    pol  ΔCC
                + ⟨Λ |ˆgN + ˆFN  + ˆGN (D    )|CC ⟩.

The subscript N indicates that the operator is normal ordered with respect to the Hartree–Fock state. Here, a polarization operator Ĝpol(D) was introduced,

                M∑  (  )⊤
ˆGpol(D ΔCC) = - 1    ˆFu   RuvFvel(D ΔCC ).
              2 uv

which depends on the elements of the difference density matrix DΔCC. These are defined as

DΔpCqC = ⟨HF |ˆa†pˆaq|CC⟩- DHFpq ,

hence, they do not depend on the Lagrangian multipliers.

17.5.2 Computational details: SCF calculations

To carry out a PE-SCF calculation with the DSCF or RIDFT module, you have to specify the following in the control file:

$point_charges pe [options]  
<length unit>  
<no. MM sites> <order k> <order pol> <length exclude list>  
<list of MM sites: exclude list, xyz coords, multipole mom., pol. tensor>

length unit
specifies the unit for the MM site coordinates (use AA or AU)
no. MM sites
the amount of MM sites (length of the list)
order k
the order of multipoles used (0: point charges, 1: dipole moments, 2: quadrupole moments, 3: octupole moments)
order pol
the treatment of polarizabilities (0: none, 1: isotropic, 2: anisotropic)
length exclude list
number of elements in the exclude list
list of MM sites
each MM sites is described on one line, entries separated by blanks; first entry is the exclude list of with as much elements as defined in the head line (If the first element in the exclusion list of one site occurs in the exclude list of another site, they do not contribute to each others polarization); next follows the MM site coordinates in (x,y,z positions), the point charge, the dipole moment (for k 1, x,y,z component), the quadrupole moment (for k 2, xx, xy, xz, yy, yz, zz component), the octupole moment (for k = 3, xxx, xxy, xxz, xyy, xyz, xzz, yyy, yyz, yzz, zzz component), the polarizability ( one component for pol-order 1, xx, xy, xz, yy, yz, zz component for pol-order 2)

An example for a polarizable embedding with coordinates given in , point charges and isotropic polarizabilities:

$point_charges pe  
6 0  1  1  
  39   -0.2765102481    2.5745845304    3.5776314866    0.038060  15.217717  
  39    1.3215071687    2.3519378014    2.8130403183   -0.009525  14.094642  
  39   -0.5595582934    1.2645007691    4.7571719292   -0.009509  14.096775  
  39   -1.5471918244    2.5316479230    2.3240961995   -0.009519  14.096312  
  39   -0.3207417883    4.1501938400    4.4162313889   -0.009507  14.096476  
  41   -1.1080691595    4.9228723099   -1.6753825535    0.038060  15.217717  
  41   -0.9775910525    6.5274614891   -2.4474576239   -0.009525  14.094642  
  41   -2.5360480539    4.8923046027   -0.6040781123   -0.009509  14.096775  
  41    0.3630448878    4.6028736791   -0.7155647205   -0.009519  14.096312  
  41   -1.2817317422    3.6689143712   -2.9344225518   -0.009507  14.096476

All values are given in atomic units (except coordinates if stated otherwise). These data are mandatory. An alternative input format can also be used by specifying daltoninp as option on the $point_charges line. The format is completely compatible with the current Dalton 2015 input format (The definition can be found in the manual from http://daltonprogram.org/www/documentation.html. See there for more information).

In addition, you can specify further options on the same line as the $point_charges flag. These are:

Limitations with respect to standard SCF computations:

The energy of a PE-SCF calculation printed in the output contains the following terms:

EPE -SCF = EQM + EQM/MM,es + Epol

Here, EQM is the energy of the quantum mechanical method of your choice, EQM∕MM,es the electrostatic interaction energy between the QM and the MM region, and Epol the energy gain due to the total of induced dipole moments. If necessary, missing terms can be computed without knowledge of the electron distribution.

At the moment, TURBOMOLE does not offer the possibility to generate the necessary potentials or to create a potential file from a set of coordinates. Embedding potentials can be obtained from literature or generated by approaches like the LoProp method. [212] Atom centered polarizabilities are also available from other methods or from experiment. Finally, there are some polarizable force fields which, in principle, can be used for the PE method (for example, the AMOEBA force field).

17.5.3 Computational details: PERI-CC2 calculations

Apart from the definition of the embedding described above, the input for PERI-CC2 calculations is the same as without polarizable embedding.

There are several limitations for the use of PERI-CC2: