Fluctuating-charge models have been used to model polarization effects in molecular mechanics methods. However, they overestimate polarizabilities in large systems. Previous attempts to remedy this have been at the expense of forbidding intermolecular charge-transfer. Here, we investigate this lack of size-extensivity and show that the neglect of terms arising from charge conservation is partly responsible; these terms are also vital for maintaining the correct translational symmetries of the dipole moment and polarizability that classical electrostatic theory requires. Also, QTPIE demonstrates linear-scaling polarizabilities when coupling the external electric field in a manner that treats its potential as a perturbation of the atomic electronegativities. Thus for the first time, we have a fluctuating-charge model that predicts size-extensive dipole polarizabilities, yet allows intermolecular charge-transfer.
Deep Dive into Size-extensive polarizabilities with intermolecular charge transfer in a fluctuating-charge model.
Fluctuating-charge models have been used to model polarization effects in molecular mechanics methods. However, they overestimate polarizabilities in large systems. Previous attempts to remedy this have been at the expense of forbidding intermolecular charge-transfer. Here, we investigate this lack of size-extensivity and show that the neglect of terms arising from charge conservation is partly responsible; these terms are also vital for maintaining the correct translational symmetries of the dipole moment and polarizability that classical electrostatic theory requires. Also, QTPIE demonstrates linear-scaling polarizabilities when coupling the external electric field in a manner that treats its potential as a perturbation of the atomic electronegativities. Thus for the first time, we have a fluctuating-charge model that predicts size-extensive dipole polarizabilities, yet allows intermolecular charge-transfer.
Polarization is an important but often neglected phenomenon in many classical molecular dynamics simulations. [1][2][3] Of the methods invented to model polarization in this context, [4,5] fluctuating-charge models provide a unified treatment of polarization and charge transfer, and thus most strongly resemble quantum-mechanical electronic structure methods. While various implementations like EEM [6,7], QEq [8,9], fluc-q [10][11][12][13][14], AACT [15][16][17], CPE [18][19][20], ABEEM [21,22], CHARMM C22 [23][24][25], EVBbased models [26][27][28], and others [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] are popular, they still suffer from some uncorrected pathologies. For example, many fluctuating-charge models are known to:
- describe physically unreasonable charge distributions for geometries far from equilibrium, [40,[49][50][51][52] 2. exhibit super-linear scaling of polarizabilities unless intermolecular charge transfer is forbidden, [15-17, 19, 53, 54] and 3. predict the wrong direction of charge transfer. [40] These problems are not due to imperfect parameters, but are symptomatic of deficiencies in the theoretical framework. Our previous work [50,51] had led us to propose the QTPIE charge model, [49,55] which has already been shown to cure the first problem for systems are electrically neutral overall. Here, we describe a solution to the last two problems as well.
In fluctuating-charge models, the electrostatic energy usually takes the form
where q i is the charge on atom i, J ij is the Coulomb interaction between atoms i and j, J ii is the chemical hardness of atom i, and v i is the electronegativity of atom i. Due to electronegativity equalization, [56,57] the solution is constrained to a total chargeQ = q !1 " q i i =1 N # . By introducing the chemical potential [58, 59] µ as a Lagrange multiplier, we construct the free energy F q, µ ( ) = E ! µ q “1 ! Q ( ) and minimize it to obtain J 1
Gaussian elimination on µ gives the analytic solution
where ! “1 = 1 T J “1 1 is the Schur complement of J in Eq. ( 2). Therefore, the electrostatic energy attains its minimum at
Interestingly, the scalar ! , which has dimensions of hardness, quantifies how much the charge constraint changes the charge distribution.
Consider models where v is a vector of atomic electronegativities, i.e. v i = ! i . We then couple such a model to an external electrostatic field ! " (where ν indexes spatial directions) by introducing the usual dipole coupling term so that the electrostatic energy becomes
Thus the effect of coupling to an electrostatic field is to transform the atomic electronegativities by
The dipole moment and polarizability tensor are then
For all systems of finite extent, ! > 0 ; however, many published results in the literature are, surprisingly, missing the terms in! . This leads to physically incorrect behavior. For example, under the change of origin R i! ! R i! + " ! , the dipole moment transforms as d ! ! d ! + " ! Q and the polarizability as ! “# ! ! “# if and only if terms in ! are included. It is therefore unnecessary to require specific choices of origin [23-25, 32, 54] to simulate translational invariance.
We now investigate the size extensivity of ( 7) and ( 8). Consider a system with n identical copies of a subsystem comprised of m atoms, with each copy separated by a distance ! " that is larger than the spatial extent of one subsystem. We use the overbar to denote quantities related to a single subsystem. The nuclear coordinates are then
) )
and v = v,…, v ( ) T . In the limit ! " # , the subsystems decouple and J becomes approximately block diagonal, with inverse
In this limit, the total dipole moment and polarizability then become
where the subsystem dipole moment and polarizability are defined analogously to ( 7) and ( 8), i.e.
where ! “1 = 1 T J “1 1 and Q = Q / n is the total charge of each identical subsystem. The second term in (11) represents the summed contributions of m point charges, each of charge Q and placed at coordinates 0, 1,…, n ! 1 ( )1 respectively. When Q = 0 , the dipole moment (11) becomes size-extensive. However, the second term in the polarizability expression (12) grows cubically with n, which is physically incorrect. Note that neither the dipole moment nor the polarizability would be size consistent if the terms in ! were neglected.
We now describe a way to obtain size-extensive dipole polarizabilities in QTPIE.
As shown earlier [52,55], any bond-space fluctuating charge model [32,60], including QTPIE, can be written in the form of (1) but with effective atomic voltages v that are not identical to the atomic electronegativities. For QTPIE, the model is defined only for Q = 0, but this is not a serious limitation in practice. The effective atomic voltages are given by
where S ij = ! i ! j is the overlap integral between atomic basis functions ! i r; R i
Instead of the usual coupling (5), we now apply the transfo
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