**Table of Contents**

As the field of computatinal chemistry matures, there is a growing demand to be able to handle
systems that are an order of magnitude (or more) larger than have previously been possible. Molecular
Mechanics (MM) has been the staple for analyzing large biochemical systems because of
its raw speed and efficiency. While useful, MM has a number of intrinsic limitations in its ability to
model chemical systems and so it would be very beneficial to be able to model these systems with
semiempirical or even *ab initio* methods. Even with the explosive growth
in computer hardware and software, applying semiempirical to enourmous systems has been
impossible due to two fundamental limitations: CPU time and memory requirements. Semiempirical
is significanly faster than *ab initio* but still inherently faces these same two
stumbling blocks.

AMPAC now supports the use of sparse matrices,
which is turned on by the keyword, SPARSE
and its ally PSOLVE.
Sparse matrices, unlike the traditional full matrices, only store and operate on those elements that are
considered non-negligable. This allows for a dramatic reduction in memory required and CPU time
utilized because for systems that are large enough, the bulk of the matrix elements can be eliminated.
This has the potential to greatly overcome the two major bottlenecks that have limited traditional
semiempirical to at most a few thousand atoms. Of course some molecular systems are just too
spatially compact, preventing us from eliminating enough matrix elements to make sparse matrices
useful. Nevertheless, there is a broad range of large molecules that are now in range that were
simply to impractical or even impossible before. For a more detailed discussion, of sparse matrices
see Chapter 7, *Sparse Matrix Methods*.

Transition states play a vital role studying chemical reactions and their mechanisms. Locating and refining a transition state is significantly more difficult than the corresponding task of searching for minima. Because of their importance, local optimizers specially suited for finding transition states and other critical points have been developed (such as TRUSTG, LTRD, and TS). However, these methods require a good guess for the transition state, which is often unavailable. Moreover, even once the transition state is located, it must be properly characterized. Thus, locating a transition state typically consists of 5 steps: (1) locate approximate transition state; (2) gradient minimization to refine the transition state; (3) computation of the frequencies; (4) removal of spurious negative eigenvalues; and (5) testing the reaction pathway. (A more detailed discussion of finding transition states is found in the section called “Characterization of Stationary Points”.)

CHN (and its predecessor CHAIN) was developed as a powerful means of
automating this search for transition states. This methods combines all five search steps into
a single unified method. Rather than working from a guess transition state, CHN attempts to
locate the transition state(s) between a given pair of minima (respectively the reactants and
the products of the chemical reaction under study). This avoids the major limitation of local
gradient minimizers and focuses the search to the specific transition state(s) of interest.
CHN forms a smooth chain of geometries stretching between starting and ending geometries,
which are then progressively refined until the transition state is located. This strategy
makes CHN very stable and reliable and helps prevent it from falling to other nearby critical
points. CHN locates only the highest
transition state along the chain of geometries. FULLCHN behaves the same as CHN except that it attempts to locate
all critical points along the chain. FULLCHN is particularly useful for analysing multiple
step reactions. A complete description of CHN and FULLCHN can be found in Chapter 8, *CHN Methods*.

One of the great challenges for modern quantum chemistry is the need to locate and characterize many geometric minima on a complex multidimensional potential energy surface. For very small well-behaved systems, characterizing the system’s potential energy surface can be accomplished by a straightforward search for minima and transition states. As chemical systems under study become larger and more complicated, the number of possible critical points increases exponentially. For many chemical systems (particularly biological systems) it is important to identify the conformation of a molecule with the lowest energy. This is a daunting challenge since there may be hundreds or even thousands of reasonable candidate structures to look for. This challenge is known as the “multiple-minima” problem.

AMPAC includes simulated annealing as tool to help tackle these
complex problems. Simulated annealing is a heuristic algorithm for finding multiple minima (or
critical points) on a potential energy surface. This is a very flexible tool, which can be
used to address a variety of related types of problems. In its simplest usage, it can be used
to automatically search a large region of the potential energy surface and report the
important structures that it finds. By applying various optimization constraints and penalty
functions, the annealing search can be focused to search for related geometries, such as the
set of conformers of a molecule. The original implementation of simulated annealing (ANNEAL) searches exclusively for minima.
This idea has been extended in AMPAC by the addition of three new
variations (MANNEAL, GANNEAL, and TSANNEAL), which search for transition states and other critical
points as well as minima. An extensive review of simulated annealing can be found in Chapter 13, *Simulated Annealing*.

All methods in AMPAC require solving for the wavefunction of the molecule. For some methods, the wavefunction must be solved for hundreds or even thousands of times in the course of the procedure (such as with reaction grids and simulated annealing). For this reason, solution of the wavefunction must be both fast and robust. AMPAC’s wavefunction solving engine satisfies both of these critera. Typical quantum chemistry programs utilize Roothan’s procedure where the Fock matrix is repeatedly diagonalized. This procedure will often fail to converge or converge very slowly for some types of systems. In AMPAC, the wavefunction is solved for as an unconstrained optimization using standard reliable optimization tools. Quadratically-convergent SCF (QCSCF) is also available to handle more difficult cases. (See the section called “The Self Consistent Field Procedure” for more details.)

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