Abstract
The potential energy surface (PES) is a central concept in computational chemistry. A PES is the relationship – mathematical or graphical – between the energy of a molecule (or a collection of molecules) and its geometry. The Born–Oppenheimer approximation says that in a molecule the nuclei are essentially stationary compared to the electrons. This is one of the cornerstones of computational chemistry because it makes the concept of molecular shape (geometry) meaningful, makes possible the concept of a PES, and simplifies the application of the Schrödinger equation to molecules by allowing us to focus on the electronic energy and add in the nuclear repulsion energy later; this third point, very important in practical molecular computations, is elaborated on in Chapter 5. Geometry optimization and transition state optimization are explained.
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- 1.
Equations marked with an asterisk are those which should be memorized.
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Henry Eyring, American chemist. Born Colonia Juarárez, Mexico, 1901. Ph.D. University of California, Berkeley, 1927. Professor Princeton, University of Utah. Known for his work on the theory of reaction rates and on potential energy surfaces. Died Salt Lake City, Utah, 1981.
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Michael Polanyi, Hungarian-British chemist, economist, and philosopher. Born Budapest 1891. Doctor of medicine 1913, Ph.D. University of Budapest, 1917. Researcher Kaiser-Wilhelm Institute, Berlin, 1920–1933. Professor of chemistry, Manchester, 1933–1948; of social studies, Manchester, 1948–1958. Professor Oxford, 1958–1976. Best known for book “Personal Knowledge”, 1958. Died Northampton, England, 1976.
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Max Born, German-British physicist. Born in Breslau (now Wroclaw, Poland), 1882, died in Göttingen, 1970. Professor Berlin, Cambridge, Edinburgh. Nobel Prize, 1954. One of the founders of quantum mechanics, originator of the probability interpretation of the (square of the) wavefunction (Chapter 4).
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J. Robert Oppenheimer, American physicist. Born in New York, 1904, died in Princeton 1967. Professor California Institute of Technology. Fermi award for nuclear research, 1963. Important contributions to nuclear physics. Director of the Manhattan Project 1943–1945. Victimized as a security risk by senator Joseph McCarthy's Un-American Activities Committee in 1954. Central figure of the eponymous PBS TV series (Oppenheimer: Sam Waterston).
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Ludwig Otto Hesse, 1811–1874, German mathematician.
References
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A standard molecular surface, corresponding to the size as determined experimentally (e.g. by X-ray diffraction) encloses about 98 per cent of the electron density. See e.g. Bader RFW, Carroll MT, Cheeseman MT, Chang, C (1987) J Am Chem Soc 109:7968
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Added in press:
Kraka E, Cremer D (2010) Review of computational approaches to the potential energy surface and some new twists, the unified reaction valley approach URVA. Acc Chem Res 43:591–601
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Appendices
Easier Questions
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1.
What is a potential energy surface (give the two viewpoints)?
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2.
Explain the difference between a relaxed PES and a rigid PES.
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3.
What is a stationary point? What kinds of stationary points are of interest to chemists, and how do they differ?
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4.
What is a reaction coordinate?
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5.
Show with a sketch why it is not correct to say that a transition state is a maximum on a PES.
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6.
What is the Born–Oppenheimer approximation, and why is it important?
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7.
Explain, for a reaction A → B, how the potential energy change on a PES is related to the enthalpy change of the reaction. What would be the problem with calculating a free energy/geometry surface?
Hint: Vibrational frequencies are normally calculated only for stationary points.
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8.
What is geometry optimization? Why is this process for transition states (often called transition state optimization) more challenging than for minima?
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9.
What is a Hessian? What uses does it have in computational chemistry?
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10.
Why is it usually good practice to calculate vibrational frequencies where practical, although this often takes considerably longer than geometry optimization?
Harder Questions
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1.
The Born–Oppenheimer principle is often said to be a prerequisite for the concept of a potential energy surface. Yet the idea of a potential energy surface (Marcelin 1915) predates the Born–Oppenheimer principle (1927). Discuss.
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2.
How high would you have to lift a mole of water for its gravitational potential energy to be equivalent to the energy needed to dissociate it completely into hydroxyl radicals and hydrogen atoms? The strength of the O–H bond is about 400 kJ mol−1; the gravitational acceleration g at the Earth’s surface (and out to hundreds of kilometres) is about 10 m s−2. What does this indicate about the role of gravity in chemistry?
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3.
If gravity plays no role in chemistry, why are vibrational frequencies different for, say, C–H and C–D bonds?
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4.
We assumed that the two bond lengths of water are equal. Must an acyclic molecule AB2 have equal A–B bond lengths? What about a cyclic molecule AB2?
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5.
Why are chemists but rarely interested in finding and characterizing second-order and higher saddle points (hilltops)?
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6.
What kind(s) of stationary points do you think a second-order saddle point connects?
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7.
If a species has one calculated frequency very close to 0 cm−1 what does that tell you about the (calculated) potential energy surface in that region?
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8.
The ZPE of many molecules is greater than the energy needed to break a bond; for example, the ZPE of hexane is about 530 kJ mol−1, while the strength of a C–C or a C–H bond is only about 400 kJ mol−1. Why then do such molecules not spontaneously decompose?
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9.
Only certain parts of a potential energy surface are chemically interesting: some regions are flat and featureless, while yet other parts rise steeply and are thus energetically inaccessible. Explain.
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10.
Consider two potential energy surfaces for the HCN HNC reaction: A, a plot of energy versus the H–C bond length, and B, a plot of energy versus the HNC angle. Recalling that HNC is the higher-energy species, sketch qualitatively the diagrams for A and B.
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Lewars, E.G. (2011). The Concept of the Potential Energy Surface. In: Computational Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3862-3_2
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DOI: https://doi.org/10.1007/978-90-481-3862-3_2
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