BioPHYSICAL chemistry

286 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

If the potential expression is subjected to a simple least-squares approach,

then the potential will decrease until it reaches one of the false minima.

Once in the false minimum, the calculated potential will never change

as that would require allowing an increase in the potential. To reach

the true minimum, the calculations have been modified to allow for an

increase in the potential for a short time. One such approach is through

a molecular-dynamics calculation in which the atoms are subject to motion

that allows the potential to temporarily increase. In these models, atoms

are free but constrained by forces. The force on the ith atom, Fi, is given by

the product of the mass, mi, and the acceleration, ai, which can be written

in terms of the position, ri, or the potential Vi:

(13.15)

The atoms are assigned velocities derived by assigning a distribution of

velocities. Typically a Maxwellian distribution is used at a certain tem-

perature (Figure 13.13):

(13.16)

This distribution has an average velocity of zero, and

equal proportions of positive and negative velocities. As the

assigned temperature is increased, the average velocity

remains zero, but the distribution broadens with a larger

percentage of higher velocities.

Once the forces and velocities are established, the system

is allowed to evolve for a short time period of about 1 ps

and the process is repeated again using the new positions.

During each time step, once the velocitiesare assigned,

atoms are allowed to move in small time steps, Δt:

(13.17)

After a specified number of time steps, the velocities are

rescaled and the potentials are recalculated. At certain

time points, the geometry may be optimized to avoid any

ΔΔ∇

Δ

xtV

t

i Ii xI total mi

=− 9 () ( )

2

2

f

m

kT

i i e

B

m

kT

ii

() BI

/

9

9

=

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

−

2

(^323)
2
2
π

LH

J

iiii J

ii

i

mm

t

V

== =−

d

d

2

2

∂

∂

VFrF

V

r

=−∫ dor=−

d

d

a

r

t

=

d

d

2

2

298 K

1000 K

0

0

Distribution

f(v

)i

Velocity vi (m/s)

Figure 13.13
Maxwellian
distributions of
velocities at 298
and 1000 K.