Previous: Part 2. Statistical Mechanics
The goal of statistical mechanics is to derive the properties of a macroscopic system based on atomic and molecular concepts. The fundamental level is quantum mechanics, but in practice, a hierarchy of approximations is used, and they are discussed in this chapter. We start with a conceptual model of classical statistical mechanics in the 19th century, then move on to quantum statistical mechanics, and finally consider the transition to the quasi-classical approximation of statistical mechanics.
The important approximation that allows us to bridge the gap between quantum and classical statistical mechanics is the Born-Oppenheimer approximation. This approximation introduces a potential energy surface that is basically equivalent to the potential energy of atoms in classical statistical mechanics. Additionally, the transformation of the electronic wave function into electronic density allows us to visualize chemical bonds.
Molecular mechanics and molecular dynamics are examples of the successful use of these ideas to solve many practical problems. However, it is important to keep in mind quantum effect corrections, which are discussed in the last section. Taken together, these approaches provide the benefits of classical statistical mechanics while maintaining an acceptable level of accuracy.
- Classical statistical mechanics
- Quantum statistical mechanics
- The Born-Oppenheimer Approximation
- Molecular Mechanics and Molecular Dynamics
- Quantum effects in the quasi-classical approximation
Classical statistical mechanics
Classical statistical mechanics completed the development of the 19th-century molecular-kinetic theory. It is assumed that the atoms obey the equations of classical mechanics. Each atom is at a given time in a specific position q with coordinates {qxi, qyi, qzi} and momentum (mass times velocity, pi = mi vi) with components {pxi, pyi, pzi}. The Hamiltonian (H) of the whole system is given as follows:
H = T + U
where T is the kinetic energy of all atoms, and U is the potential energy that describes the interaction between atoms. The motion of atoms in the entire system is governed by a system of Hamilton’s equations, and below only the equations of motion for the x-coordinate of the i-th atom are written:
The number of atoms in a macroscopic system, such as a candle flame, is on the order of Avogadro’s number (NA = 6.02⋅1023), which is extremely large. It is impossible to write down a complete system of equations, not speaking of solving it. However, the equations above provide a conceptual model of 19th-century atomism, which can be visualized by replacing the point masses with small interacting balls. Let us look at a figure from Wikipedia (Computational chemistry, Molecular dynamics) that shows the current configuration of water molecules in molecular dynamics simulation:
There is a video in Wiki that shows the changing positions of water molecules; the calculation is based on Hamilton’s equations. There are strong interactions between the atoms in the water molecule and weaker intermolecular interactions between molecules. However, water also has hydrogen bonds between the water molecules, which contribute to the unique properties of water.
In classical statistical mechanics, all interactions between atoms should be supplied externally; in other words, they must be determined experimentally. In any case, the picture and video in the Wiki should not be treated literally. It is only an approximation that leads to good results in certain cases, but the balls and rods do not reflect reality.
Quantum statistical mechanics
Nowadays we know that atoms are made of electrons and nuclei, and that chemical reactions, such as the burning of a candle, involve the redistribution of electrons between nuclei. This process cannot be described using classical mechanics; it requires the use of quantum mechanics. At the level of a macroscopic system, we need statistical quantum mechanics.
In this case, the wave function of the whole system is required, which in the coordinate representation depends on the coordinates of all nuclei (the vector R includes all components of the coordinates of all nuclei) and the coordinates of all electrons (the vector r includes all components of the coordinates of all electrons): Ψ(r, R). In statistical quantum mechanics, only the energy spectrum of the system is employed, that can be found by solving the stationary Schrödinger equation (H is the quantum Hamiltonian, E is the energy):
Ψ H(r, R) = E Ψ(r, R)
The energy values (E0, E1, E2, …) are called the eigenvalues of the quantum Hamiltonian, and the wave functions (Ψ0, Ψ1, Ψ2, …) are called the eigenvectors in the Hilbert space. For a macro system, it is impossible to write out this equation in detail, let alone solve it. Most important that we can no longer talk about the state of the system in the spirit of classical mechanics. The relationship between the wave function and experiments is determined by the Born rule, which states that the square of the wave function is equal to the probability of observing that configuration. Classical mechanics is deterministic, while quantum mechanics deals with probabilities.
The Born-Oppenheimer Approximation
To compute macroscopic properties, only the energy spectrum is required, so the discussion of the wave function comes down to the energy spectrum of the entire system. However, the representation of a macroscopic system as an energy spectrum is too abstract, and it would be good to have at least some form of the visual images of classical statistical mechanics. The Born-Oppenheimer approximation is an important step towards achieving this goal. It involves factoring the total wave function into two components:
Ψ(r, R) = Ψel(r, R)Ψnuc(R)
The mass of the nuclei is much larger than the mass of the electrons, so it is assumed possible to separate the motion of the nuclei, described by the wave function Ψnuc(R), from the motion of the electrons. The coordinates of the nuclei R play the role of a parameter in the electronic wave function Ψel(r, R), that is, this wave function is related to this configuration of nuclei. The substitution of the factorization into the original Schrödinger equation results in the problem being split into two steps. First, the electronic Schrödinger equation is to be solved:
Hel Ψel(r, R) = Еel(R) Ψel(r, R)
The difference from the previous equation is the role of the coordinates of the nuclei. Now they are fixed before finding the solution of the electronic Schrödinger equation and thus the electronic energy Eel(R) is a function of the coordinates of the nuclei. Let us have a look at the picture with a potential energy surface from Wiki (Potential energy surface):
The figure shows the electronic energy for the H-O-H water molecule as a function of two geometric parameters: the H-O distance (assumed to be the same for both bonds) and the angle between the bonds. To obtain the potential energy surface, the electronic Schrödinger equation is solved many times for different geometries of the molecule. The minimum on the potential energy surface represents the equilibrium configuration of the water molecule.
Additionally one can transform the many-electron wave function, which depends on the coordinates of all electrons, into the electron density, which is a function of only three spatial coordinates. In Wiki there is a picture of the electron density of aniline (Electron density):
The figure shows contours of the electron density at different values when the nuclei are fixed. Large values of the electron density on the left show the position of the nuclei, medium values allow visualization of the chemical bonds, and small values of the electron density on the right show the boundaries of the aniline molecule.
Thus, the solution of the electronic Schrödinger equation for a given configuration of nuclei corresponds to the potential energy of the atoms in the classical Hamiltonian. Thus, the Born-Oppenheimer approximation paves the way for a visual representation of classical statistical mechanics with molecules moving in space, as was shown in the case of water. The movement of atoms in the classical Hamiltonian is associated with the rearrangement of the electronic density, and the electronic energy determines the potential energy of the atomic interactions.
Molecular Mechanics and Molecular Dynamics
Numerical solution of the Schrödinger electron equation is a time-consuming task, and the number of electrons that can be included is limited. As a result, there are methods to model intramolecular and intermolecular interactions in the form of fitted analytical functions, when the parameters needed are determined from quantum chemical calculations and available spectroscopic information. This approach is known as molecular mechanics, as it is based on the classical concept of potential energy between atoms.
In molecular mechanics, a specific set of parameters obtained to describe atom interactions is called a force field. There are different force fields designed for different tasks and kind of molecules, and for practical calculations it is necessary to choose one of them. A typical task in molecular mechanics is to find the equilibrium geometries of molecular structures, which can then be used in molecular dynamics.
In molecular dynamics, the equations of motion of the classical Hamiltonian are solved numerically (see the video with the motion of water molecules in Wiki: Molecular dynamics). In this case, the configuration of molecules with a specified force field models a small part of the macrosystem. To avoid the influence of surface effects, periodic boundary conditions are used — an atom that goes beyond the geometric boundaries of the simulation box appears on the other side.
The equations of motion of the classical Hamiltonian are time-symmetric, that is, integrating the system backward in time from a final configuration should lead to the initial configuration. We return to this problem in the third chapter during a discussion of the arrow of time, since this property of the Hamiltonian conflicts with the Clausius inequality.
In numerical molecular dynamics algorithms, there are rounding errors. Combined with the instability of the equations of motion in classical mechanics (dynamic chaos), they result in the fact that reversing the integration in time in molecular dynamics does not lead to the original system. Thus, numerical rounding errors provide the arrow of time in molecular dynamics.
Quantum effects in the quasi-classical approximation
Remember that the Born-Oppenheimer approximation does not mean a way back to classical statistical mechanics. Let us consider monatomic inert gases. Formally, a molecule in this case consists of a single atom, and therefore there are no internal degrees of freedom, leaving only the translational motion of individual atoms. However, even in this case, corrections to classical statistical mechanics are needed due to the uncertainty principle and the indistinguishability of particles in quantum mechanics. Without them it would be impossible to obtain the correct entropy value.
Thus, the Born-Oppenheimer approximation only paves the way for a transition to a quasi-classical approximation, in which the visual representation of atomic motion is preserved to some extent. For example, we can think of the candle combustion as the motion of nuclei, when the potential energy changes due to the electron density redistribution. However, even the motion of inert gas atoms at the level of classical statistical mechanics should not be taken literally, let alone for more complex processes.
The potential energy surface above has been considered only for the ground electronic state. At the same time, the solution of the electron Schrödinger equation leads to many electron states (Eel,0, Eel,1, Eel,2, …), i.e., in addition to the ground electronic state, there are excited electronic states, each of which has its own potential energy surface. In many cases, electronic transitions are essential and there is no classical mechanical equivalent.
In the candle combustion, the Born-Oppenheimer approximation usually inapplicable to consider chemical reactions, as the configuration of the reactant and product nuclei may correspond to different electronic states. In this case, special corrections are required to account for the interaction between the motion of the nuclei and the electrons.
Also, the second part of solution of the Schrödinger equation should not be forgotten. After the electronic Schrödinger equation, we should solve the part for the motion of nuclei in the potential energy surface:
Hnuc Ψnuc(R) = Еnuc Ψnuc(R)
Under normal conditions, one cannot neglect the quantization of the vibrational motions of nuclei. It was the quantization of vibrations that solved the problem with heat capacities of gases in classical statistical mechanics. This is another reason why the representation of the atomic movement along trajectories in molecular dynamics cannot be taken literally — the vibrational motion require quantization.
Next: Chapter 2. Equilibrium Statistical Mechanics
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