2.17. Finite Electric Fields

Electric fields can have significant influences on the electronic structure of molecules. In general, when an electric field is applied to a molecule, the electron cloud of the molecule will polarize along the direction of the field. The redistribution of charges across the molecule will then influence the wavefunction of the molecule. Even when polarization effects are not significant, the electric field still exerts a drag on the negatively and positively charged atoms of the molecule in opposite directions, and therefore affect the orientation and structure of the molecule. The combination of electrostatic and polarization effects make electric fields a useful degree of freedom in tuning e.g. reactivities, molecular structures and spectra [175]. Meanwhile, the energy/dipole moment/quadrupole moment changes of the system in the presence of small dipolar or quadrupolar electric fields are useful for calculating many electric properties of the system via numerical differentiation, including the dipole moment, quadrupole moment, dipole-dipole polarizability, quadrupole-quadrupole polarizability, etc. Such finite difference property calculations can be conveniently done using compound scripts in the ORCA Compound Scripts Repository.

An overview of relevant keywords is given in Table 2.59.

2.17.1. Dipolar Electric Fields

A uniform or equivalently speaking, dipolar electric field can be added to a calculation via the EField keyword in the %scf block:

%scf
  EField 0.1, 0.0, 0.0 # x, y, z components (in au) of the electric field
end

Although the keyword is in the %scf block, it applies the electric field to all other methods (post-HF methods, multireference methods, TDDFT, etc.) as well. Analytic gradient contributions of the electric field are available for all supported methods that already support analytic gradients, but analytic Hessian contributions are not.

Warning

The electric field functionality is not available for GFN-xTB and force field methods (as well as any method that involves GFN-xTB or force fields, e.g. QM/XTB and QM/MM! Combination with these method will result in an abort.

The sign convention of the electric field is chosen in the following way: suppose that the electric field is generated by a positive charge in the negative z direction, and a negative charge in the positive z direction, then the z component of the electric field is positive. This convention is consistent with most but not all other programs [175], so care must be taken when comparing the results of ORCA with other programs.

Another important aspect is the gauge origin of the electric field. The gauge origin of the electric field is the point (or more accurately speaking, one of the points - as there are infinitely many such points) where the electric potential due to the electric field is zero. Different choices of the gauge origin do not affect the geometry and wavefunction of the molecule, as they do not change the electric field felt by the molecule, but they do change the energy of the molecule. The default gauge origin is the (0,0,0) point of the Cartesian coordinate system, but it is possible to choose other gauge origins via the EFieldOrigin keyword in the %scf block.

%scf
 EFieldOrigin CenterOfMass      # use center of mass
              CenterOfNucCharge # use center of nuclear charge
              0.0, 0.0, 0.0     # use given X,Y,Z as origin (default: 0,0,0)
                                # in the units chosen for the coordinates (Angstrom/Bohr)
end

Note the default gauge origin of the electric field is different from the default gauge origin of the ELPROP module, which is the center of mass. If the user chooses the center of mass/nuclear charge as the gauge origin of the electric field, the gauge origin will move as the molecule translates; this has important consequences. For example, in an MD simulation of a charged molecule in an electric field, the molecule will not accelerate, unlike when EFieldOrigin is fixed at a given set of coordinates, where the molecule will accelerate forever. In general, CenterOfMass and CenterOfNucCharge are mostly suited for the finite difference calculation of electric properties, where one frequently wants to choose the center of mass or nuclear charge as the gauge origin of the resulting multipole moment or polarizability tensor. Instead, a fixed origin is expected to be more useful for simulating the changes of wavefunction, geometry, reactivity, spectra etc. under an externally applied electric field, as experimentally the electric field is usually applied in the lab frame, rather than the comoving frame of the molecule.

2.17.2. Quadrupolar Electric Fields

Similar to dipolar electric fields, quadrupolar fields can be added via the QField keyword in the %scf block:

%scf
  QField 0.1, 0.0, 0.0, 0.05, 0.0, 0.0 # xx, yy, zz, xy, xz, yz components (in au)
                                       # of the quadrupolar field
end

The gauge origin of the quadrupolar field is the same as that of the dipolar electric field.

2.17.3. Combination of Dipolar and Quadrupolar Electric Fields

Dipolar and quadrupolar electric fields can be combined using the respective EField and QField keywords in the %scf block. This allows one to simulate a gradually varying electric field, for example the following input specifies an electric field that has a strength of 0.01 au at the gauge origin ((0,0,0) by default), pointing to the positive z direction, and increases by 0.001 au for every Bohr as one goes in the positive z direction:

%scf
  EField 0.0, 0.0, 0.01
  QField 0.0, 0.0, 0.001, 0.0, 0.0, 0.0
end

As a second example, one can also simulate an ion trap:

%scf
  QField -0.01, -0.01, -0.01, 0.0, 0.0, 0.0
end

Under this quadrupolar field setting, a particle will feel an electric field that points towards the gauge origin, whose strength (in au) is 0.01 times the distance to the gauge origin (in Bohr). This will keep cations close to the origin, but pushes anions away from the origin. Unfortunately, there is no analytic gradient available for quadrupolar fields.

Notes on Electric Fields

• An au (atomic unit) is a fairly large unit for electric fields: 1 au = 51.4 V/Angstrom. By comparison, charged residues in proteins, as well as scanning tunneling microscope (STM) tips, typically generate electric fields within about 1 V/Angstrom; electrode surfaces usually generate electric fields within 0.1 V/Angstrom under typical electrolysis conditions [175]. If the molecule is not close to the source of the electric field, it is even harder to generate strong electric fields: for example, a 100 V voltage across two metal plates that are 1 mm apart generates an electric field of merely \(10^{-5}\) V/Angstrom. Therefore, if experimentally a certain strength of homogeneous electric field seems to promote a reaction, but no such effect is found in calculation, please consider the possibility that the experimentally observed reactivity is due to a strong local electric field near the electrode surface (that is much higher than the average field strength in the system), or due to other effects such as electrolysis. Conversely, if you predict a certain molecular property change at an electric field strength of, e.g. \(>\) 0.1 au, it may be a non-trivial question whether such an electric field can be easily realized experimentally.

• The electric field breaks the rotational symmetry of the molecule, in the sense that rotating the molecule can change its energy. Therefore, geometry optimizations in electric fields cannot be done with internal coordinates. When the user requests geometry optimization, the program automatically switches to Cartesian coordinates if it detects an electric field. While Cartesian coordinates allow the correct treatment of molecular rotation, they generally lead to poor convergence, so a large number of iterations is frequently necessary. Also, since certain geometry optimization tasks (notably transition state optimization) can currently only be done under internal coordinates, electric fields cannot be used in such types of optimization tasks yet.

• Similarly, when the molecule is charged, its energy is not invariant with respect to translations. However, when there is only a dipolar electric field but no other translational symmetry-breaking forces (quadrupolar field, point charges, wall potentials), a charged molecule will accelerate forever in the field, and its position will never converge. Therefore, for geometry optimizations within a purely dipolar electric field and no wall potentials, we do not allow global translations of the molecule, even when translation can reduce its energy. For MD simulations we however do allow the global translations of the molecule by default. If this is not desired, one can fix the center of mass in the MD run using the CenterCOM keyword (section Run).

• For frequency calculations in electric fields, we do not project out the translational and rotational contributions of the Hessian (equivalent to setting ProjectTR false in %freq; see Vibrational Frequencies for details). Therefore, the frequencies of translational and rotational modes can be different from zero, and can mix with the vibrational modes. When the electric field is extremely small but not zero, the “true” translational/rotational symmetry breaking of the Hessian may be smaller than the symmetry breaking due to numerical error; this must be kept in mind when comparing the frequency results under small electric fields versus under zero electric field (in the latter case ProjectTR is by default true). Besides, when the translational and rotational frequencies exceed CutOffFreq (which is 1 cm\(^{-1}\) by default; see section Vibrational Frequencies), their thermochemical contributions are calculated as if they are vibrations.

• While the program allows the combination of electric fields with an implicit solvation model, the results must be interpreted with caution, because the solvent medium does not feel the electric field. The results may therefore differ substantially from those given by experimental setups where both the solute and the solvent are subjected to the electric field. If the solvent’s response to the electric field is important, one should use an explicit solvation model instead. Alternatively, one can also simulate the electric field in the implicit solvent by adding inert ions (e.g. Na\(^+\), Cl\(^-\)) to the system. Similarly, implicit solvation models cannot describe the formation of electrical double layers in the electric field and their influence on solute properties, so in case electrical double layers are important, MD simulations with explicit treatment of the ions must be carried out.

• The electric field not only contributes to the core Hamiltonian, but has extra contributions in GIAO calculations, due to the magnetic field derivatives of dipole integrals. In the case of a dipolar electric field, the GIAO contributions have been implemented, making it possible to study e.g. the effect of electric fields on NMR shieldings, and as a special case, nucleus independent chemical shieldings (NICSs), which are useful tools for analyzing aromaticity. Quadrupolar fields cannot be used in GIAO calculations at the moment.

2.17.4. Keywords

Table 2.59 %scf block input keywords and options for finite electric fields.

Keyword	Option	Description
EField	<x, y, z>	Activates dipolar electric field with x, y, z components (in au)
QField	<xx, yy, zz, xz, xz, yz>	Activates quadrupolar electric field with xx, yy, zz, xz, xz, yz components (in au)
EFieldOrigin	CenterOfMass	Sets origin to center of mass
	CenterOfNucCharge	Sets origin to center of nuclear charge
	<X, Y, Z>	Sets origin to X,Y,Z coordinates (default: 0,0,0) in the units chosen for the coordinates (Angstrom/Bohr)