Editing Computational chemistry (section)

==== Chemical dynamics ====
After the electronic and [[molecular geometry|nuclear]] variables are [[separation of variables|separated]] within the Born–Oppenheimer representation), the [[wave packet]] corresponding to the nuclear [[degrees of freedom (physics and chemistry)|degrees of freedom]] is propagated via the [[time evolution]] [[operator (physics)]] associated to the time-dependent [[Schrödinger equation]] (for the full [[molecular Hamiltonian]]).<ref>{{Cite journal |last=Butler |first=Laurie J. |date=October 1998 |title=Chemical Reaction Dynamics Beyond the Born-Oppenheimer Approximation |url=https://www.annualreviews.org/doi/10.1146/annurev.physchem.49.1.125 |journal=Annual Review of Physical Chemistry |language=en |volume=49 |issue=1 |pages=125–171 |bibcode=1998ARPC...49..125B |doi=10.1146/annurev.physchem.49.1.125 |issn=0066-426X |pmid=15012427}}</ref>  In the [[complementarity (physics)|complementary]] energy-dependent approach, the time-independent [[Schrödinger equation]] is solved using the [[scattering theory]] formalism. The potential representing the interatomic interaction is given by the [[potential energy surface]]s. In general, the [[potential energy surface]]s are coupled via the [[vibronic coupling]] terms.<ref>{{Cite journal |last1=Ito |first1=Kenichi |last2=Nakamura |first2=Shu |date=June 2010 |title=Time-dependent scattering theory for Schrödinger operators on scattering manifolds |url=http://doi.wiley.com/10.1112/jlms/jdq018 |journal=Journal of the London Mathematical Society |language=en |volume=81 |issue=3 |pages=774–792 |arxiv=0810.1575 |doi=10.1112/jlms/jdq018 |s2cid=8115409}}</ref>

The most popular methods for propagating the [[wave packet]] associated to the [[molecular geometry]] are:
* the [[Chebyshev polynomials|Chebyshev (real) polynomial]],<ref>{{Cite journal |last1=Ambrose |first1=D |last2=Counsell |first2=J. F |last3=Davenport |first3=A. J |date=1970-03-01 |title=The use of Chebyshev polynomials for the representation of vapour pressures between the triple point and the critical point |url=https://dx.doi.org/10.1016/0021-9614%2870%2990093-5 |journal=The Journal of Chemical Thermodynamics |volume=2 |issue=2 |pages=283–294 |doi=10.1016/0021-9614(70)90093-5 |bibcode=1970JChTh...2..283A |issn=0021-9614}}</ref>
* the [[multi-configuration time-dependent Hartree]] method (MCTDH),<ref>{{Cite journal |last1=Manthe |first1=U. |last2=Meyer |first2=H.-D. |last3=Cederbaum |first3=L. S. |date=1992-09-01 |title=Wave-packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl |url=https://doi.org/10.1063/1.463007 |journal=The Journal of Chemical Physics |volume=97 |issue=5 |pages=3199–3213 |bibcode=1992JChPh..97.3199M |doi=10.1063/1.463007 |issn=0021-9606}}</ref>
* the semiclassical method
* and the split operator technique explained below.<ref name="Lukassen-2018">{{Cite journal |last1=Lukassen |first1=Axel Ariaan |last2=Kiehl |first2=Martin |date=2018-12-15 |title=Operator splitting for chemical reaction systems with fast chemistry |journal=Journal of Computational and Applied Mathematics |volume=344 |pages=495–511 |doi=10.1016/j.cam.2018.06.001 |issn=0377-0427 |s2cid=49612142 |doi-access=free}}</ref>

===== Split operator technique =====
How a computational method solves quantum equations impacts the accuracy and efficiency of the method. The split operator technique is one of these methods for solving differential equations. In computational chemistry, split operator technique reduces computational costs of simulating chemical systems. Computational costs are about how much time it takes for computers to calculate these chemical systems, as it can take days for more complex systems. Quantum systems are difficult and time-consuming to solve for humans. Split operator methods help computers calculate these systems quickly by solving the sub problems in a quantum [[differential equation]]. The method does this by separating the differential equation into two different equations, like when there are more than two operators. Once solved, the split equations are combined into one equation again to give an easily calculable solution.<ref name="Lukassen-2018" />

This method is used in many fields that require solving differential equations, such as [[Mathematical biology|biology]]. However, the technique comes with a splitting error. For example, with the following solution for a differential equation.<ref name="Lukassen-2018" />

<math display="inline">e^{h(A+B)} </math>

The equation can be split, but the solutions will not be exact, only similar. This is an example of first order splitting.<ref name="Lukassen-2018" />

<math display="inline">e^{h(A+B)} \approx e^{hA}e^{hB} </math>

There are ways to reduce this error, which include taking an average of two split equations.<ref name="Lukassen-2018" />

Another way to increase accuracy is to use higher order splitting. Usually, second order splitting is the most that is done because higher order splitting requires much more time to calculate and is not worth the cost. Higher order methods become too difficult to implement, and are not useful for solving differential equations despite the higher accuracy.<ref name="Lukassen-2018" />

Computational chemists spend much time making systems calculated with split operator technique more accurate while minimizing the computational cost. Calculating methods  is a massive challenge for many chemists trying to simulate molecules or chemical environments.<ref name="Lukassen-2018" />