On the Cartesian Representation of the Molecular Polarizability Tensor Surface by Polynomial Fitting to<i>Ab Initio</i>Data

Omodemi, Oluwaseun; Sprouse, Sarah; Herbert, Destyni; Kaledin, Martina; Kaledin, Alexey L.

doi:10.1021/acs.jctc.1c01015

Cited by 5 publications

(18 citation statements)

References 61 publications

(114 reference statements)

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“…With

α_{p} (boldr)

being a function of the geometry, the commonly applied Thole's “dipole smearing” model (see below), 10,12,13,22 derived for constant atomic polarizabilities to eliminate these singularities is theoretically not valid. Recently, however, it was suggested to avoid constructing

({boldA}^{- 1} + T)

directly and approximate its inverse using a weighted power series

{((), A^{- 1} goodbreak+ boldT)}^{- 1} ≅ A - λ_{2} (boldr) ATA + λ_{3} (boldr) ATATA - λ_{4} (boldr) ATATATA + \dots

with

λ_{n} (boldr)

being configuration dependent short‐range scalar correction factors that tend to 1 at long range 6 and with

λ_{1} \equiv 1

. The 3

\times

3 molecular polarizability tensor may now be constructed using Equation (),

α_{NL}^{(L)} ((), r) = bold1 \sum_{p} α_{p} ((), r) - λ_{2} ((), r) \sum_{p \neq q} α_{p} ((), r) α_{q} ((), r) T_{pq} + λ_{3} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{s \neq p, q} α_{s} ((), r) T_{ps} T_{}

…”

Section: Methodsmentioning

confidence: 99%

“…Recently, however, it was suggested to avoid constructing

({boldA}^{- 1} + T)

directly and approximate its inverse using a weighted power series

{((), A^{- 1} goodbreak+ boldT)}^{- 1} ≅ A - λ_{2} (boldr) ATA + λ_{3} (boldr) ATATA - λ_{4} (boldr) ATATATA + \dots

with

λ_{n} (boldr)

being configuration dependent short‐range scalar correction factors that tend to 1 at long range 6 and with

λ_{1} \equiv 1

. The 3

\times

3 molecular polarizability tensor may now be constructed using Equation (),

α_{NL}^{(L)} ((), r) = bold1 \sum_{p} α_{p} ((), r) - λ_{2} ((), r) \sum_{p \neq q} α_{p} ((), r) α_{q} ((), r) T_{pq} + λ_{3} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{s \neq p, q} α_{s} ((), r) T_{ps} T_{sq} - λ_{4} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{\begin{matrix} s ((), \neq p) \\ s^{'} \end{matrix}}

…”

Section: Methodsmentioning

confidence: 99%

“…terms in Equations (3 and 4) contain the products

c_{n} c_{1}

c_{n} c_{1}^{2}

c_{n} c_{1}^{3}

, and so on for n = 2, 3, …, L in addition to the linear terms in

c_{n}

. As such, the least‐squares problem of fitting Equation () to training data must be solved using non‐linear regression techniques, for example, function minimization 4–6 …”

Section: Methodsmentioning

confidence: 99%

“…Recent attention to the analytic representation of the static dipole polarizability, using various robust fitting methods, has revealed a rapidly growing field in the simulation of Raman spectra of molecules and liquids using ab initio quality potential energy (PES) and polarizability tensor (PTS) surfaces. To note, several computational studies of molecular PTSs employing methods ranging from many‐body expansions and (deep) neural networks (NN) to permutationally invariant polynomials (PIP) and others have been reported in the past few years 1–6 . Formal expression for a PTS may depend on the specific problem at hand, as for instance has been done for a number of diatomics 7–9 and simple polyatomics such as H 2 O and thus‐derived (H 2 O) m systems, 1–3,10 or may be generic in nature and constructed using translational‐rotational symmetry preserving matrix products of “transition dipole”‐like quantities, as has been originally demonstrated for larger polyatomics and liquid water systems 4,5 .…”

Section: Introductionmentioning

confidence: 99%

“…Formal expression for a PTS may depend on the specific problem at hand, as for instance has been done for a number of diatomics 7–9 and simple polyatomics such as H 2 O and thus‐derived (H 2 O) m systems, 1–3,10 or may be generic in nature and constructed using translational‐rotational symmetry preserving matrix products of “transition dipole”‐like quantities, as has been originally demonstrated for larger polyatomics and liquid water systems 4,5 . Another symmetry‐adapted generic approach introduced independently more recently is to approximately invert a linear response “relay” matrix 11–13 using perturbation theory, as was examined for a series of small polyatomics 6 . Notably, both of the generic approaches described in the literature involve polarizability formulas that necessarily depend on the fitting parameters, that is, polynomial expansion coefficients, in a non‐linear way.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Permutationally invariant polynomial representation of polarizability tensor surfaces for linear regression analysis

Omodemi

Kaledin

2022

J Comput Chem

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A linearly parameterized functional form for a Cartesian representation of molecular dipole polarizability tensor surfaces (PTS) is described. The proposed expression for the PTS is a linearization of the recently reported power series ansatz of the original Applequist model, which by construction is non-linear in parameter space. This new approach possesses (i) a unique solution to the least-squares fitting problem; (ii) a low level of the computational complexity of the resulting linear regression procedure, comparable to those of the potential energy and dipole moment surfaces; and (iii) a competitive level of accuracy compared to the non-linear PTS model. Calculations of CH 4 PTS, with polarizabilities fitted to 9000 training set points with the energies up to 14,000 cm À1 show an impressive level of accuracy of the linear PTS model

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“…With

α_{p} (boldr)

({boldA}^{- 1} + T)

directly and approximate its inverse using a weighted power series

{((), A^{- 1} goodbreak+ boldT)}^{- 1} ≅ A - λ_{2} (boldr) ATA + λ_{3} (boldr) ATATA - λ_{4} (boldr) ATATATA + \dots

with

λ_{n} (boldr)

being configuration dependent short‐range scalar correction factors that tend to 1 at long range 6 and with

λ_{1} \equiv 1

. The 3

\times

3 molecular polarizability tensor may now be constructed using Equation (),

α_{NL}^{(L)} ((), r) = bold1 \sum_{p} α_{p} ((), r) - λ_{2} ((), r) \sum_{p \neq q} α_{p} ((), r) α_{q} ((), r) T_{pq} + λ_{3} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{s \neq p, q} α_{s} ((), r) T_{ps} T_{}

…”

Section: Methodsmentioning

confidence: 99%

“…Recently, however, it was suggested to avoid constructing

({boldA}^{- 1} + T)

directly and approximate its inverse using a weighted power series

{((), A^{- 1} goodbreak+ boldT)}^{- 1} ≅ A - λ_{2} (boldr) ATA + λ_{3} (boldr) ATATA - λ_{4} (boldr) ATATATA + \dots

with

λ_{n} (boldr)

being configuration dependent short‐range scalar correction factors that tend to 1 at long range 6 and with

λ_{1} \equiv 1

. The 3

\times

3 molecular polarizability tensor may now be constructed using Equation (),

α_{NL}^{(L)} ((), r) = bold1 \sum_{p} α_{p} ((), r) - λ_{2} ((), r) \sum_{p \neq q} α_{p} ((), r) α_{q} ((), r) T_{pq} + λ_{3} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{s \neq p, q} α_{s} ((), r) T_{ps} T_{sq} - λ_{4} ((), r) \sum_{p, q} α_{p} ((), r) α_{q} ((), r) \sum_{\begin{matrix} s ((), \neq p) \\ s^{'} \end{matrix}}

…”

Section: Methodsmentioning

confidence: 99%

“…terms in Equations (3 and 4) contain the products

c_{n} c_{1}

c_{n} c_{1}^{2}

c_{n} c_{1}^{3}

, and so on for n = 2, 3, …, L in addition to the linear terms in

c_{n}

. As such, the least‐squares problem of fitting Equation () to training data must be solved using non‐linear regression techniques, for example, function minimization 4–6 …”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Permutationally invariant polynomial representation of polarizability tensor surfaces for linear regression analysis

Omodemi

Kaledin

2022

J Comput Chem

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Revisiting theN(N + 1)/2‐sites‐type Gaussian charge model for permutationally invariant polynomial fitting of global molecular tensor surfaces

Kudratov

Kaledin

2023

Int J of Quantum Chemistry

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The exact expressions for the dipole, quadrupole, and octupoles of a collection of N point charges involve summations of corresponding tensors over the N sites weighted by their charge magnitudes. When the point charges are atoms (in a molecule) the N‐site formula is an approximation, and one must integrate over the electron density to recover the exact multipoles. In the present work we revisit the N(N + 1)/2‐site point charge density model of Hall (Chem. Phys. Lett. 6, 501, 1973) for the purpose of fitting ab initio derived multipole moment hypersurfaces using permutationally invariant polynomials (PIP). We examine new approaches in PIP‐fitting procedures for the dipole, quadrupole, octupole moments, and polarizability tensor surfaces (DMS, QMS, OMS and PTS, respectively) for a non‐polar CCl4 and a polar CHCl3 and show that compared to the primitive N‐site model the N(N + 1)/2‐site model appreciably improves the relative RMSE of the DMS and does much more substantially so, by an order of magnitude, for the corresponding ones of QMS and OMS. Training datasets are obtained by sampling potential energies up to 18 000 cm−1 above the global minima, generated by molecular dynamics simulations at the DFT B3LYP/aug‐cc‐pVDZ level of theory.

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Fitting Potential Energy Surfaces by Learning the Charge Density Matrix with Permutationally Invariant Polynomials

Hashem,

Foust,

Kaledin

et al. 2023

J. Chem. Theory Comput.

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The electronic energy in the Hartree−Fock (HF) theory is the trace of the product of the charge density matrix (CDM) with the one-electron and twoelectron matrices represented in an atomic orbital basis, where the two-electron matrix is also a function of the same CDM. In this work, we examine a formalism of analytic representation of a generic molecular potential energy surface (PES) as a sum of a linearly parameterized HF and a correction term, the latter formally representing the electron correlation energy, also linearly parameterized, by expressing the elements of CDM using permutationally invariant polynomials (PIPs). We show on a variety of numerical examples, ranging from exemplary two-electron systems HeH + and H 3 + to the more challenging cases of methanium (CH 5 + ) fragmentation and high-energy tautomerization of formamide to formimidic acid that such a formulation requires significantly fewer, 10−20% of PIPs, to accomplish the same accuracy of the fit as the conventional representation at practically the same computational cost.

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On the Cartesian Representation of the Molecular Polarizability Tensor Surface by Polynomial Fitting toAb InitioData

Cited by 5 publications

References 61 publications

Permutationally invariant polynomial representation of polarizability tensor surfaces for linear regression analysis

Permutationally invariant polynomial representation of polarizability tensor surfaces for linear regression analysis

Revisiting theN(N + 1)/2‐sites‐type Gaussian charge model for permutationally invariant polynomial fitting of global molecular tensor surfaces

Fitting Potential Energy Surfaces by Learning the Charge Density Matrix with Permutationally Invariant Polynomials

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