Chapter 8 – Theoretical Biophysics  327

The term on the right is the determinant of the spin–​orbit matrix. The χ elements of the

spin–​orbit matrix are the wave functions for the various orthogonal (i.e., independent) wave

functions for the n individual particles.

In some cases of light atomic nuclei, such as those of hydrogen, QM MD can model the

effects of the atomic nuclei as well as the electronic orbitals. A similar HF method can be

used to approximate the total wave function; however, since nuclei are composed of bosons

as opposed to fermions, the spin–​orbit matrix permanent is used instead of the determinant

(a matrix permanent is identical to a matrix determinant, but instead the signs in front of the

matrix element product permutations being a mixture of positive and negative as is the case

for the determinant, they are all positive for the permanent). This approach has been valuable

for probing effects such as the quantum tunneling of hydrogen atoms, a good example being

the mechanism of operation of an enzyme called “alcohol dehydrogenase,” which is found in

the liver and requires the transfer of an atom of hydrogen using quantum tunneling for its

biological function.

KEY POINT 8.2

The number of calculations for n atoms in classical MD is ~O(n²) for pairwise potentials,

though if a cutoff is employed, this reduces to ~O(n). If next-​nearest-​neighbor effects

(ormore) are considered, ~O(n^α) calculations are required where α ≥ 3. Particle–​mesh

methods can reduce the number of calculations to ~O(n log n) for classical pairwise

interaction models. The number of direct calculations required for pairwise interactions

in QM MD is ~O(n³), but approximations to the total wave function can reduce this to

~O(n^2.7).

Ab initio force fields are not derived from predetermined potential energy functions

since they do not assume preset bonding arrangements between the atoms. They are

thus able to model chemical bond making and breaking explicitly. Different bond

coordination and hybridization states for bond making and breaking can be mod­

eled using other potentials such as the Brenner potential and the reactive force field

(ReaxFF) potential.

The use of semiempirical potentials, also known as tight-​binding potentials, can

reduce the computational demands in ab initio simulations. A semiempirical potential

combines the fully QM-​based interatomic potential energy derived from ab initio mod­

eling using the spin–​orbit matrix representation described earlier. However, the matrix

elements are found by applying empirical interatomic potentials to estimate the overlap

of specific atomic orbitals. The spin–​orbit matrix is then diagonalized to determine

the occupancy level of each atomic orbital, and the empirical formulations are used to

determine the energy contributions from these occupied orbitals. Tight-​binding models

include the embedded-​atom method also known as the tight-​binding second-​moment

approximation and Finnis–​Sinclair model and also include approaches that can approxi­

mate the potential energy in heterogeneous systems including metal components such as

the Streitz–​Mintmire model.

Also, there are hybrid classical and quantum mechanical methods (hybrid QM/​MM).

Here, for example, classical MD simulations may be used for most of a structure, but the

region of the structure where most fine spatial precision is essential is simulated using QM-​

derived force fields. These can be applied to relatively large systems containing ~10⁴–​10⁵

atoms, but the simulation time is limited by QM and thus restricted to simulation times of

~10–​100 ps. Examples of this include the simulation of ligand binding in a small pocket of

a larger structure and the investigation of the mechanisms of catalytic activity at a specific

active site of an enzyme.