Structural and Thermodynamic Studies of Synuclein Proteins Related to Parkinson’s Disease: - Metallic Systems: A Quantum Chemist's Perspective - page 47

aggregation.of.α-synuclein..The.principle.and.application.of.CGMD.(coarse.grained.
MD).simulations.can.be.found.elsewhere. 57
The. total. energy. for. a. QM/MM. system. can. be. represented. as. a. sum. of. three.
contributions:

E E = + E + E . (2.17)

.

QM MM QM MM /

where
E MM . represents. the. energy. calculated. using. a. force. ield. (e.g.,. CHARMM,. as.
described.earlier)
E QM .represents.the.energy.of.the.part.of.the.system.treated.using.quantum.mechan-
ics,.usually.described.within.the.Born-Oppenheimer.approximation 58 .as

electrons electron nuclei electron electr ons nuclein nuclein

1
2

Z
R

1 Z Z

R

∑ ∑ ∑ ∑ ∑ ∑ ∑

2 j i j

E = − ∇ − + + . (2.18)

QM

r

ij ij ij

.

i i j i j < i i j

The.four.terms.in.Equation.2.18.represent.contribution.from.the.kinetic.energy,.the.
electron-nucleus. interaction,. the. electron-electron. interaction,. and. the. nuclear-
nuclear.repulsion.interaction,.respectively..When.coupled.with.molecular.mechanics,.
additional.van.der.Waals.and.electrostatic.force.terms.are.added.for.each.QM.atom..
Thus,.the.energy.of.the.combined.QM/MM.system,. E QM/MM ,.is.written

QM electrons MM atoms QM nuc leis MM atoms QM nucleis MM atoms

⎛
⎜

⎞
⎟

q
r

Z q
R

A
R

B
R

∑ ∑ ∑ ∑ ∑ ∑

j i j ij ij

E = − + + −

QM MM /

12 6

ij ij ij ij

.

i j i j i j

.
(2.19)

where
A ij .and. B ij .are.van.der.Waals.parameters
Z ij .is.the.effective.charge.of.nuclei. i
q j .is.the.atomic.partial.charge.of.atom. j
r ij .represents.distance.between.electron. i .and.atom. j
R ij .represents.the.distance.between.nuclei. i .and.atom. j

The. calculation. of. E MM . is. performed. as. described. earlier. for. empirical. force.
ields..Standard.methods.in.ab.initio.quantum.mechanics.(such.as.the.Hartree-Fock.
method).and.in.DFT.(via.the.Kohn-Sham.equations) 59 .are.used.to.calculate.the. E QM ..
The.so-called.“semiempirical”.QM.methods,.which.are,.in.principle,.less.accurate.
than.QM.methods.but.also.less.expensive.to.evaluate,.may.also.be.used.for.the.QM.
region.
One.critical.consideration.in.a.combined.QM/MM.method.is.the.means.by.which.
the. QM. and. MM. parts. are. joined.. Various. strategies. have. been. proposed. to. treat.
the.boundary.atoms.between.the.QM.and.MM.part.to.make.the.calculation.of.the.
coupling,. E QM/MM ,.more.accurate.and.eficient..If.there.is.a.covalent.bond.between.