Quantum Calculations in Solution for Large to Very Large ... · even if globally neutral, has charges in some parts of the molecule. In conclusion, it is not possible to simulate

Quantum Calculations in Solution for Large to VeryLarge Molecules: A New Linear Scaling QM/Continuum

Approach

Yvon Maday

Laboratoire Jacques-Louis Lions - UPMC, Paris, FranceIUF

and Division of Applied Maths Brown University, Providence USA

Conca60

Yvon Maday (LJLL - UPMC/ Brown Univ) Solvation QM/MM Conca’s 60th 1 / 69

Introduction

Preamble . . .

Interdisciplinary is important for Carlos, his involvement in applied maths,in the university, in direction to interdisciplinary industrial applications isan example.

Multiscale analysis has been an important focus of his research

I will thus speak of multiscale approaches in computational chemistry


Introduction

Preamble . . .





Introduction

Preamble . . .





Introduction

Multiscale metods in computational chemistry . . .

The importance of multiscale modeling in computational chemistry hasbeen strongly attested by the Nobel Prize in Chemistry to Karplus, Levittand Warshel.

Within this frame, a molecular system is divided into smaller subsystems,each of which being treated using different methods. The most prominentexample of a multiscale model is the combined quantum mechanics andmolecular mechanics (QM/MM) method.


Introduction





Introduction





Introduction

Multiscale metods . . .environment

One of the most important aspects of embedding models is the couplingbetween the central part and the environment. This coupling can bedivided into three subclasses : mechanical embedding, electrostaticembedding and polarizable embedding

mechanical embedding scheme is performed on a purely classical leveland it is therefore only suitable for groundstate energy calculations.

electrostatic embedding scheme includes one-electron operators in theelectronic Hamiltonian. This directly affects the electron density ofthe central part and thus also the calculated molecular properties.The environment is normally represented by atomic partial charges ormultipole moments.

polarizable embedding scheme is currently the most advancedQM/MM type embedding scheme. Here, the polarization effects inthe environment are also taken into account.


Introduction

Multiscale metods . . .environment

One of the most important aspects of embedding models is the couplingbetween the central part and the environment. This coupling can bedivided into three subclasses : mechanical embedding, electrostaticembedding and polarizable embedding

mechanical embedding scheme is performed on a purely classical leveland it is therefore only suitable for groundstate energy calculations.

electrostatic embedding scheme includes one-electron operators in theelectronic Hamiltonian. This directly affects the electron density ofthe central part and thus also the calculated molecular properties.The environment is normally represented by atomic partial charges ormultipole moments.

polarizable embedding scheme is currently the most advancedQM/MM type embedding scheme. Here, the polarization effects inthe environment are also taken into account.


Introduction

Solvation effects . . .

90% of the chemistry is performed in solvation. There is a reciprocalinfluence of the solvant of the molecule under investigation

Especially for proteins containing many (partial) charges, atoms in theclassical region can contribute to the properties of the quantum regionthrough polarization interactions up to a distance of 20 Angstrom 1

This comes from the long range interaction due to the fact that a protein,even if globally neutral, has charges in some parts of the molecule. Inconclusion, it is not possible to simulate the protein and the surroundingwater molecule in such a large region, especially if you take into accountthe fact that the configuration richness should then be properly scanned inorder to get averages.

1. Beerepoot, M. T. ; Steindal, A. H. ; Ruud, K. ; Olsen, J. M. H. ; Kongsted, J. Com-putational and Theoretical Chemistry 2014, 1040-1041, 304311.


Introduction







Introduction







Introduction







Introduction


Continuum solvation models are nowadays among the most popular toolsin computational chemistry to include the effects of the chemicalenvironment in the description of a molecular property or process.

These methods are based on the electrostatic problem of a density ofcharge accommodated in a properly shaped cavity surrounded by auniform, dielectric continuum.


Introduction


Continuum solvation models are nowadays among the most popular toolsin computational chemistry to include the effects of the chemicalenvironment in the description of a molecular property or process.

These methods are based on the electrostatic problem of a density ofcharge accommodated in a properly shaped cavity surrounded by auniform, dielectric continuum.


Introduction


Molecule in solution : solvation problem.


Introduction


The strength of these solvation models is their simplicity of use andcost-effectiveness, which has granted them a prominent role in a vast fieldof application, ranging from chemistry to biophysics to materials science.Continuum solvation models have been used together with both classicalmolecular mechanics (MM), quantum mechanics (QM) and hybrid(QM/MM) levels of theory to describe the solute and several differentformulations and implementations exist, including fully polarizableQM/MM/Continuum ones.


Introduction


The electrostatic energy of the charge distribution ρ (classical pointcharges, electric dipoles and multipoles in force-field models, classicalnuclear charges and quantum electronic charge density in first-principle orsemi-empirical models) carried by the solute is modified by the presence ofthe solvent, and an extra term, called the electrostatic contribution to thesolvation energy, and denoted here by E s, must be added to theelectrostatic energy computed in vacuo. The contribution E s can bewritten as

E s =1

2

∫R3

ρ(r)V r(r) dr,

where V r is the reaction-field potential generated by the chargedistribution ρ in presence of the dielectric continuum.


Introduction

Molecule in solution : COnductor-like Screening MOdel.

In the COSMO model, the electrostatic contribution to the solvationenergy is given by

E sC =

1

2f (εs)

∫Ωρ(r)W (r) dr,

where f (εs) = εs−1εs+k is an empirical function of εs (k is a paramater taken

equal to 0.5 in COSMO), and where W is the solution to the boundaryvalue problem

−∆W= 0 in Ω,W= −Φ on Γ.

(1)


Introduction


In the COSMO model, the electrostatic contribution to the solvationenergy is given by

E sC =

1

2f (εs)

∫Ωρ(r)W (r) dr,

where f (εs) = εs−1εs+k is an empirical function of εs (k is a paramater taken

equal to 0.5 in COSMO), and where W is the solution to the boundaryvalue problem

−∆W= 0 in Ω,W= −Φ on Γ.

(2)


Introduction


An example of domain Ω

Cafein molecule


Introduction


Another example of domain Ω

Peridin clorophill protein molecule


Introduction


The difficulty thus does not come from the PDE itself but from thecomplexity of the geometry.


Introduction


The usual method to compute E sC is to represent W by a single layer

potential :

∀r ∈ Ω := Ω ∪ Γ, W (r) =

∫Γ

σC(s)

|r − s|ds,

where the surface charge density σ is obtained by solving

∀s ∈ Γ, (SΓσC) (s) = −Φ(s). (3)

where

∀s ∈ Γ, (SΓσ)(s) =

∫Γ

σ(s′)

|s− s′|ds′,


Introduction



Introduction



Introduction



Introduction


Cances, M, Stamm

Figure: Iterations history with respect to number of Spherical Harmonics


Introduction


Cances, Lagardere, Lipparini,M. Mennucci, Stamm

Ideal Scaling

Parallel Implementation

Figure: Iterations history


Introduction


CSC = various Continous Surface Charge approaches


Introduction


CSC = various Continous Surface Charge approaches


Introduction

Continous surface charge approaches versus DD-Cosmo.


Introduction

More . . .

At the same time this bottleneck is solved, we have also investigated thecentral problem of electronic structure calculations from a numericalanalysis point of view.

This means a priori analysis


Introduction

More . . .


This means a priori analysis


Introduction

A priori analysis . . .

over 10,000 papers a year . . .very few in mathsover 15% of the resources in scientific computing centerstwo Nobel prizesTough problem ..

high dimensional / high complexity

Coulombic interaction

Antisymetry of the wave function : electrons are fermions Pauliprinciple

energy of individual atoms versus energy difference : 108 scale


Introduction








Introduction








Introduction








Introduction


Nonlinear elliptic eigenvalue problem resulting from the minimization of anenergy under a normalization constraint, e.g. Kohn Sham approach : basedon the theorem of Hohenberg-KohnIn density functional theory, the total energy of a system is expressed as afunctional of the charge density ρ such that

∫ρ = N as

E [ρ] = Ts [ρ] +

∫vextρ+ VH [ρ] + Exc [ρ]


Introduction


E [ρ] = Ts [ρ] +

∫vextρ+ VH [ρ] + Exc [ρ]

where Ts is the KohnSham kinetic energy which is expressed in terms ofthe KohnSham orbitals φi as

Ts [ρ] =N∑i=1

∫(∇φi )2

vext is the external potential acting on the interacting system (atminimum, for a molecular system, the electron-nuclei interaction), VH isthe Hartree (or Coulomb) energy,

VH =

∫ ∫ρ(r)ρ(r ′)

|r − r ′|and Exc is the exchange-correlation energy. This term, is the onlyunknowns in the KohnSham approach to density functional theory.


Introduction

A priori estimatesThe main result is the following (Cances, Chakir, M. 2012 M2AN).

Theorem

Let Φ0 be a local minimizer of the Kohn-Sham problem. Then there existsr0 > 0 and N0

c such that for Nc ≥ N0c , there exists a unique discrete local

minimizer Φ0Nc

in the set

ΦNc ∈ VNNc∩MΦ0 | ‖ΦNc − Φ0‖H1

#≤ r0

. If

we assume either that eLDAxc ∈ C [m]([0,+∞)) or that ρc + ρ0 > 0 on Γ,

then we have the following estimates :

‖Φ0Nc− Φ0‖Hs

#≤ Cs,εN

−(m−s+1/2−ε)c , (4)

|ε0i ,Nc− ε0

i | ≤ CεN−(2m−1−ε)c , (5)

γ‖Φ0Nc− Φ0‖2

H1#≤ IKS

Nc− IKS ≤ C‖Φ0

Nc− Φ0‖2

H1#, (6)

for all −m + 3/2 < s < m + 1/2 and ε > 0, and for some constants γ > 0,Cs,ε ≥ 0, Cε ≥ 0 and C ≥ 0, where the ε0

i ,Nc’s are the eigenvalues of the

symmetric matrix Λ0Nc, the Lagrange multiplier of the matrix constraint∫

Γ φi ,Ncφj ,Nc = δij .Yvon Maday (LJLL - UPMC/ Brown Univ) Solvation QM/MM Conca’s 60th 28 / 69

Introduction

More . . .


This means a priori analysis . . .a posteriori analysis


Introduction

A posteriori Analysis

We started at the beginning of this century to work on this subject withGabriel Turinici, in order to certify the discretization error for theapproximation of one major model used in computational quantumchemistry : the Hartree-Fock equations.

Latter some contributions in the litterature (A. Zhou team) on the aposteriori estimate for discretization errors and the conclusion leading tomesh adaptation

Now we want to go further taking more into account the challenges of thesimulation that does not only bears on discretization methods . . .there arealso algorithms


Introduction






Introduction






Introduction






Introduction

Gross-Pitaevskii problem — a posteriori

In order to understand, we have first considered the plane wavediscretization of Gross-Pitaevskii nonlinear eigenvalue problem . . .with G.Dusson

−∆u + Vu + µu3 = λu avec ‖u‖L2 = 1


Introduction

Two types of errorsThe eigenvalue problem being nonlinear, the solution procedure is iterative

The two sources of errors are thus : the discretization parameter N andthe number of iterations that are performed


Introduction

Two types of errorsThe eigenvalue problem being nonlinear, the solution procedure is iterative

The two sources of errors are thus : the discretization parameter N andthe number of iterations that are performed


Introduction

Iterative processThe algorithm used to solve the equation numerically in the space XN isthe following. Starting from a given couple (u0

N , λ0N), we solve at each step

the linear equation

−∆uk∗N + VNuk∗N + (uk−1

N )2uk∗N = λk−1N uk−1

N (7)

where VN is the approximate potential i.e. the projection of the potentialV on the space XN . We find uk∗N which is a non-normalized vector. So wenormalize it and write

ukN =uk∗N‖uk∗N ‖L2

(8)


Introduction

Iterative processFinally, we define the eigenvalue as a Rayleigh quotient being

λkN =

∫Ω

(∇uk∗N )2 +

∫ΩVN(uk∗N )2 +

∫Ω

(uk∗N )4∫Ω

(uk∗N )2(9)

which is also

λkN =

∫Ω

(∇ukN)2 +

∫ΩVN(ukN)2 +

∫Ω

(ukN)4 (10)

We can check numerically that such an algorithm converges


Introduction

Errors in NWe define the two residues :

‖RkN‖L2 = ‖−∆ukN + VukN + (ukN)3 − λkNukN‖L2

(error due to the number of iterations)

‖Rk,k−1N ‖H−1

#= ‖−∆ukN + VNu

kN + (uk−1

N )2ukN − λk−1N uk−1

N ‖H−1#

(error due to the discretization dimension)The a posteriori analysis bounds the H1 error by the sum of twocomponents :

errk =1

N‖Rk

N‖L2N

+ ‖ukN − uk−1N ‖L2 + |λkN − λk−1

N |+ |1− 1

‖uk?N ‖L2

|

errN = ‖Rk,k−1N ‖H−1 +

1

N‖V − VN‖L2‖ukN‖H1

And the total error is :

errtotal ≤ errk + errN


Introduction

Errors in N


Introduction

Errors in k


Introduction

Balancing the Errors in N and k potential k4


Introduction

Balancing the Errors in N and k potential k2


Introduction

Balancing the Errors in N and k


Introduction

We are now generalizing this approach to Kohn-Sham with E. Cances, G.Dusson, B. Stamm, M. Vohralik.


Introduction

More . . .


This means a priori analysis . . .a posteriori analysis . . .analysis of the model


Introduction

and beyond . . .

We want now to incorporate the error due to the model . . .

Indeed, what is of interest for us is the solution to the full, original,Schrodinger equation. What is the link between Schrodinger and one ofthe feasible model.

Kohn Sham, DFT

Hartree Fock

?


Introduction

and beyond . . .



Kohn Sham, DFT

Hartree Fock

?


Introduction

and beyond . . .



Kohn Sham, DFT

Hartree Fock

?


Introduction

and beyond . . .



Kohn Sham, DFT

Hartree Fock

?


Introduction

and beyond . . .



Kohn Sham, DFT ? ?

Hartree Fock


Introduction

and beyond . . .



Kohn Sham, DFT better correlation models

Hartree Fock


Introduction

and beyond . . .



Kohn Sham, DFT better correlation models

Hartree Fock post Hartree Fock methods


Introduction

Schrodinger’s equation : simplification

The Schrodinger’s problem is too large ! !

For electronic structure calculations ψ(x1, x2, . . . , xN) 2 : two approaches

Hartree Fock approximation : ψ(x1, x2, . . . , xN) is a Slatterdeterminant = det(ϕi (xj))

Density functional : ρ(x) =∫ψ2(x , x2, . . . , xN)dx2 . . . dxN

The first one is suggested from the Pauli (antisymetry) principle, thesecond comes from Hohenberg and Kohn’s theorem and Kohn et Sham’sanzatz telling that the behavior of the whole system at fundamental stateis determined by the only knowledge of the density : that is a function inR3.

2. parametrized by the position of the nucleiYvon Maday (LJLL - UPMC/ Brown Univ) Solvation QM/MM Conca’s 60th 47 / 69

Introduction








Introduction








Introduction




Hartree Fock approximation : ψ(x1, x2, . . . , xN) is a Slatterdeterminant = det(ϕi (xj)) . . .and post Hartree Fock methods




Introduction


These ideas lead to two progresses :

We are now looking for functions with variable in R3

The problem is stated as the minimization of an energy under∫ψ2 = 1 constraint

The difficulty is that the linearity of the (too large) Schrordinger problemis replaced by a highly nonlinear problem . . .involving a large number ofiterations


Introduction







Introduction







Introduction

post Hartree-Fock . . .CI-Full CI

We are using Slatter determinants to minimize the Schrodinger energy. . .leads to the following equation

Find (Φ,λ) ∈ Y × RN such that, ∀ψ ∈ Y

12

∫R3 ∇ϕi∇ψi +

∫R3 V ϕiψi + 2

∑Nj=1

∫R3

∫R3|ϕj (y)|2ϕi (x)ψi (x)

|x−y| dx dy

−∑N

j=1

∫R3

∫R3

ϕi (y)ϕj (y)ϕj (x)ψi (x)|x−y| dx dy

= λi∫R3 ϕiψi


Introduction


We are using Slatter determinants to minimize the Schrodinger energy. . .leads to the following equation

Find (Φ,λ) ∈ Y × RN such that, ∀ψ ∈ Y

12

∫R3 ∇ϕi∇ψi +

∫R3 V ϕiψi + 2

∑Nj=1

∫R3

∫R3|ϕj (y)|2ϕi (x)ψi (x)

|x−y| dx dy

−∑N

j=1

∫R3

∫R3

ϕi (y)ϕj (y)ϕj (x)ψi (x)|x−y| dx dy

= λi∫R3 ϕiψi


Introduction


We are using Slatter determinants. . .

This has led us to an eigenvalue problem . . .where we have withdrawn**only** the N lowest eigenvalues : the occupied orbitals.

There are N − N to be used : the excited states.The basic Hartree Fock determinant is written as

Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1

and we denote it asΨ0 := Ψ[1, 2, . . . ,N]

for obvious reasons as it involves the N occupied orbitals.


Introduction


We are using Slatter determinants. . .

This has led us to an eigenvalue problem . . .where we have withdrawn**only** the N lowest eigenvalues : the occupied orbitals.

There are N − N to be used : the excited states.The basic Hartree Fock determinant is written as

Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1

and we denote it asΨ0 := Ψ[1, 2, . . . ,N]

for obvious reasons as it involves the N occupied orbitals.


Introduction


The basic Hartree Fock determinant is written as

Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1 := Ψ[1, 2, . . . ,N].

Single excited determinant can then be constructed as

Ψaj := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . ,N]

where the occupied orbital j is replaced by the unoccupied orbital a.Analogously, doubly excited determinants are constructed as

Ψa,bj ,k := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . , k − 1, b, k + 1, . . . ,N]

Higher excitations involve index

µ =

(a1 . . . ak`1 . . . `k

)where ai designates an index of unoccupied orbital that replaces theoccupied one `i associated to an excitation of order k . Such an exciteddeterminant is denoted as Ψµ = XµΨ0 where Xµ is a k-order excitationoperator.Yvon Maday (LJLL - UPMC/ Brown Univ) Solvation QM/MM Conca’s 60th 52 / 69

Introduction



Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1 := Ψ[1, 2, . . . ,N].


Ψaj := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . ,N]


Ψa,bj ,k := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . , k − 1, b, k + 1, . . . ,N]


µ =

(a1 . . . ak`1 . . . `k


Introduction



Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1 := Ψ[1, 2, . . . ,N].


Ψaj := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . ,N]


Ψa,bj ,k := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . , k − 1, b, k + 1, . . . ,N]


µ =

(a1 . . . ak`1 . . . `k


Introduction



Ψ0(x) =1√N!

det(ϕi (xj))Ni ,j=1 := Ψ[1, 2, . . . ,N].


Ψaj := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . ,N]


Ψa,bj ,k := Ψ[1, 2, . . . , j − 1, a, j + 1, . . . , k − 1, b, k + 1, . . . ,N]


µ =

(a1 . . . ak`1 . . . `k


Introduction


The linear combinaison of all these excited determinants represents all theantisymetric functions that can be built . . .Actually, all the exciteddeterminants may not be so useful . . .meaning that the coefficients infront of some of these, in the expansion of the ground state solution toSchrodinger problem, may be VERY small.

Would we know this a priori, we would look for minimization on reducedexpansions based on only those that are useful.

This is the spirit of Coupled Cluster approximations


Introduction






Introduction






Introduction






Introduction






Introduction

post Hartree-Fock . . .CC

Coupled Cluster approximations

Two problems

size consistency

Which determinants are important ?


Introduction



Two problems

size consistency



Introduction



Two problems

size consistency AB = A + B . . .



Introduction



Two problems

size consistency AB = A + B . . .

Which determinants are important ? CCSD, CCSDT . . .


Introduction


Coupled Cluster approximationsAbout size consistency AB = A + B

Answer : exponential framework ...

The approximate solution is not sought as a linear combination of excitedSlater determinants but as nonlinear combination through an exponentialansatz.


Introduction






Introduction






Introduction


About the second problem CCSD, CCSDT . . .

means that we choose A PRIORI that only single, double, triple excitationare interesting . . .more excited states being useless

There are still too many possibilities CCSDT leads to N8 complexity


Introduction






Introduction






Introduction


There are still too many possibilities CCSDT leads to N8 complexityin a series of papers of Frank Neese and collaborators, it is explained howto select those few excitations that are the more relevant and onlyoptimize the coefficients in front of these.Notion of “Pair Natural Orbitals” further translated into “local ProjectedAtomic Orbitals”Linear scaling is now possible on these CC approaches with single, doubleand even triple excitations being able to capture as much as 99,8% of theenergy.

Nevertheless, as stated by Neese this involves an “incredible difficultbookkeeping problem” requiring to “look far enough ahead into thecalculation to figure out which integral is needed when and where — andwhy”.


Introduction





Introduction





Introduction



Introduction


The natural question is then to understand the link between the norm of tand the energy norm of the wave function. Following Reinhold Schneiderthis is provided by the quantity ‖t‖V defined by

‖t‖2V =

∑µ∈J

εµ|tµ|2

where εµ =∑k

i=1 λai − λ`i , and the λ’s are Hartree Fock eigenvalues inincreasing order. This norm is equivalent to the H1 norm of Ψ. The correctevaluation of the norm of the residual f(t) := (fµ(t))µ is thus

‖f‖2V ′ =

∑µ∈J

ε−1µ |fµ|2


Introduction

An adaptive strategy for Coupled Cluster Approximations

Starting from an initial index set J0 composed say of single excitations.The procedure — as usual in the adaptive process — follows the rule

ESTIMATE −→ MARK −→ REFINE :

at step i , in order to define Ji+1 we estimate those fµ(ti ) that may beadded in order to improve the accuracy of the computation.

∀µ ∈ J , fµ(ti ) :=< Ψµ|e−TiHeTi |Ψ0 >= 0

By marking those indices associated with the those that have the largestcontribution in the above V ′ (dual)-norm (i.e. with the relative weightε−1µ ) we add them in the set Ji to get a Ji+1 (finer) adapted set and we

continue recursively by enriching up to a level where the error estimator issmall at the required accuracy.


Introduction









Introduction









Introduction

More . . .


This means a priori analysis . . .a posteriori analysis . . .analysis of themodel . . .propose new approaches


Introduction

New approaches . . .

Two grids methods . . .with E. Cances, R. Chakir and L. He

Select the proper basis set :reduced basis approximation [E. Cances, C. Le Bris, YM, N. C. Nguyen, A.T. Patera, and G. Pau], [YM and U. Razafison]


Introduction

Conclusion...

There is a lot still to be done . . .some other directions

Molecular dynamics = Hamiltonian system ==> parallelization intime (with Baffico and Zerah) and (Legoll Lelievre)

Quantic control (with Salomon, Turinici)


Introduction

Conclusion...





Introduction

Conclusion...





Introduction

Thanks. . .

. . .Thanks . . .

In collaboration withEric Cances, Rachida Chakir, Genevieve Dusson, Michael J. Frisch, LouisLagardere, Filippo Lipparini, Benedetta Mennucci, Jean-Philip Piquemal,Giovanni Scalmani, Benjamin Stamm, Martin Vohralik

Supported by :

France-Berkeley Fund

ANR Manif,

CALSIMLAB ANR-11-IDEX-0004-02


Introduction

Thanks. . .

. . .Thanks . . .

In collaboration withEric Cances, Rachida Chakir, Genevieve Dusson, Michael J. Frisch, LouisLagardere, Filippo Lipparini, Benedetta Mennucci, Jean-Philip Piquemal,Giovanni Scalmani, Benjamin Stamm, Martin Vohralik

Supported by :

France-Berkeley Fund

ANR Manif,

CALSIMLAB ANR-11-IDEX-0004-02


Introduction

SPECIAL MESSAGE FROM LJLL RESEARCHERS . . .Happy BirthdayCarlo . . .

Please come more often to Paris and visit us . . .life is more fun and vibrantwith you around


Introduction




Introduction




Introduction


Introduction

Questions ? ?


Documents

Quantum Calculations in Solution for Large to Very Large ... · even if globally neutral, has charges in some parts of the molecule. In conclusion, it is not possible to simulate