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勢能面，表示某一微觀體系的勢能和相關參數（通常為原子坐標）之間的函數關係，是勢能函數的圖像。勢能面用一個或更多的坐標去表示，當用一個坐標去表示時，勢能面通常被稱為「勢能曲線」。

勢能面概念被用在物理以及化學領域, 尤其是它們的理論研究分支。 勢能面可以被用來從理論層面理解由原子組成的物質的性質, 例如：搜尋分子的最低能量構形或者計算化學反應速率。

勢能面類似於對地形的描述：對於一個有兩個自由度的體系（例如：一個鍵長、一個鍵角），體系的勢能可以類比為地形的高度，兩個自由度可以類比為描述某位置的坐標。通過這樣的描述，體系勢能隨坐標的變化可以很直觀地被表示出來。

水分子勢能面: 勢能最低點對應優化後的水分子結構， O-H 鍵長 0.0958nm， H-O-H尖角為 104.5°

A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; if there is only one coordinate, the surface is called a potential energy curve or energy profile. An example is the Morse/Long-range potential.

It is helpful to use the analogy of a landscape: for a system with two degrees of freedom (e.g. two bond lengths), the value of the energy (analogy: the height of the land) is a function of two bond lengths (analogy: the coordinates of the position on the ground).[1]


The PES concept finds application in fields such as chemistry and physics, especially in the theoretical sub-branches of these subjects. It can be used to theoretically explore properties of structures composed of atoms, for example, finding the minimum energy shape of a molecule or computing the rates of a chemical reaction.

Mathematical definition and computation 編輯

原子的空間位置通過向量 r來表示，向量的元素代表某一個原子的坐標。 向量 r可以是原子 笛卡爾坐標的集合， 也可以是原子間距離、鍵角以及二面角等.

給定一個 r，得到能量 E(r)， E(r) 是 r的函數。類比地形的描述, E 是"能量地形"的高度， 所以 E 被稱為一個面。

利用勢能面作為原子位置的函數研究化學反應時，需要計算每一個關注的分子構象能量。計算特定原子排布能量可以採用計算化學方法，本節將討論 E(r)的近似表達 。

對於一些非常簡單的化學體系或者原子間相互作用簡化處理後了的體系，給出原子坐標與能量直接函數的解析形式是可能的。一個簡單的例子是用 London-Eyring-Polanyi-Sato 勢函數[2][3][4] 描述雙氫體系中的H-H距離。

對於更複雜的系統，計算較大範圍的勢能面十分耗時。為解決上述問題，計算過程中通常選取部分結構來計算勢能面，並用計算得到的數據點插值獲得更大的勢能面。機器學習方法也可以用來構建勢能面。

應用 編輯

勢能面是概念工具協助分析分子結構與進行化學反應動力學模擬。勢能面上的點可以根據能量的一階和二階導數進行區分，駐點（梯度為0的點）對應著一個穩定的化學結構。鞍點對應著過渡態，是反應路徑上的最高點，反應路徑是連接產物與生成物的最低能量路徑。

A PES is a conceptual tool for aiding the analysis of molecular geometry and chemical reaction dynamics. Once the necessary points are evaluated on a PES, the points can be classified according to the first and second derivatives of the energy with respect to position, which respectively are the gradient and the curvature. Stationary points (or points with a zero gradient) have physical meaning: energy minima correspond to physically stable chemical species and saddle points correspond to transition states, the highest energy point on the reaction coordinate (which is the lowest energy pathway connecting a chemical reactant to a chemical product).

Attractive and repulsive surfaces 編輯

Potential energy surfaces for chemical reactions can be classified as attractive or repulsive by comparing the extensions of the bond lengths in the activated complex relative to those of the reactants and products.[5][6] For a reaction of type A + B—C → A—B + C, the bond length extension for the newly formed A—B bond is defined as R*AB = RAB − R0AB, where RAB is the A—B bond length in the transition state and R0AB in the product molecule. Similarly for the bond which is broken in the reaction, R*BC = RBC − R0BC, where R0BC refers to the reactant molecule.[7]

For exothermic reactions, a PES is classified as attractive (or early-downhill) if R*AB > R*BC, so that the transition state is reached while the reactants are approaching each other. After the transition state, the A—B bond length continues to decrease, so that much of the liberated reaction energy is converted into vibrational energy of the A—B bond.[7][8] An example is the harpoon reaction K + Br2 → K—Br + Br, in which the initial long-range attraction of the reactants leads to an activated complex resembling K+•••Br−•••Br.[7] The vibrationally excited populations of product molecules can be detected by infrared chemiluminescence.[9][10]

In contrast the PES for the reaction H + Cl2 → HCl + Cl is repulsive (or late-downhill) because R*HCl < R*ClCl and the transition state is reached when the products are separating.[7][8] For this reaction in which the atom A (here H) is lighter than B and C, the reaction energy is released primarily as translational kinetic energy of the products.[7] For a reaction such as F + H2 → HF + H in which atom A is heavier than B and C, there is mixed energy release, both vibrational and translational, even though the PES is repulsive.[7]

For endothermic reactions, the type of surface determines the type of energy which is most effective in bringing about reaction. Translational energy of the reactants is most effective at inducing reactions with an attractive surface, while vibrational excitation is more effective for reactions with a repulsive surface.[7] As an example of the latter case, the reaction F + HCl(v=1)[11] → Cl + HF is about five times faster than F + HCl(v=0) → Cl + HF for the same total energy of HCl.[12]

History 編輯

The concept of a potential energy surface for chemical reactions was first suggested by the French physicist René Marcelin in 1913.[13] The first semi-empirical calculation of a potential energy surface was proposed for the H + H2 reaction by Henry Eyring and Michael Polanyi in 1931. Eyring used potential energy surfaces to calculate reaction rate constants in the transition state theory in 1935.

See also 編輯

References 編輯

  1. ^ Potential-energy (reaction) surface in Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997)
  2. ^ Sato, S. A New Method of Drawing the Potential Energy Surface. Bulletin of the Chemical Society of Japan. 1955, 28 (7): 450. doi:10.1246/bcsj.28.450. On a New Method of Drawing the Potential Energy Surface. The Journal of Chemical Physics. 1955, 23 (3): 592. Bibcode:1955JChPh..23..592S. doi:10.1063/1.1742043. 
  3. ^ Keith J. Laidler, Chemical Kinetics (3rd ed., Harper & Row 1987) p.68-70 ISBN 0-06-043862-2
  4. ^ Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1998) p.201-2 ISBN 0-13-737123-3
  5. ^ Attractive potential-energy surface in Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997)
  6. ^ Repulsive potential-energy surface in Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997)
  7. ^ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 Keith J. Laidler, Chemical Kinetics (3rd ed., Harper & Row 1987) p.461-8 ISBN 0-06-043862-2
  8. ^ 8.0 8.1 Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1998) p.272-4 ISBN 0-13-737123-3
  9. ^ Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1998) p.263 ISBN 0-13-737123-3
  10. ^ Atkins P. and de Paula J. Physical Chemistry (8th ed., W.H.Freeman 2006) p.886 ISBN 0-7167-8759-8
  11. ^ Here v is the vibratonal quantum number.
  12. ^ Atkins P. and de Paula J. Physical Chemistry (8th ed., W.H.Freeman 2006) p.889-890 ISBN 0-7167-8759-8
  13. ^ Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics Errol G. Lewars, 2nd ed. (Springer 2011) p.21 ISBN 978-9048138616
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