Relación DDT y NC

Before molecular dynamics simulations are discussed in detail, it is worth considering a couple of the many assumptions made when a molecular dynamics simulation is performed. The validity of the results depends on the validity of these.

65 1.4.2.1.1. Ergodicity

The probability distribution of the possible states of a system is known as its ensemble. In a simulation, the system is usually heated up to the desired temperature using a constant volume simulation. Once equilibrated at the correct temperature, more natural, constant pressure simulation begins. (The volume of the system is allowed to vary (p82).) In the constant volume simulation, the NVT (constant particle number, volume and temperature) or canonical ensemble is being sampled. In the constant pressure simulation, the NPT (constant particle number, pressure and temperature) or isothermal–isobaric ensemble is being sampled.

The ensemble average of a property is the value of the property in each possible microstate of the system, weighted by the probability of the system being in that microstate.

〈𝐴〉 = 𝐴𝑖𝑃𝑖 1.16

Where 〈𝐴〉 is the ensemble average, A is the property, P is the probability and i is the microstate. The probability of a compound being found in each microstate depends on the energy of that microstate and the temperature (p91).

In molecular dynamics simulations, systems are modelled over time and the averages obtained from them are time-averages.; they are only estimates of the true ensemble averages165(p17).

In the context of molecular dynamics simulations, the ergodic hypothesis states that, given

sufficient time, a simulation will sample all of the possible states of the system and therefore that averages over the frames in a simulation reflect the ensemble average166(p307).

In practice, the ergodic assumption is seldom completely fulfilled165(p17) because the transitions between some stable states have a low probability due to the high energy of the intervening microstates. As a result, sampling may be incomplete, causing time-averages to be weighted towards the initial state of the system.

1.4.2.1.2. The Born-Oppenheimer Approximation

The mass of an atomic nucleus is thousands of times greater than the mass of each of its electrons. Consequently, when the velocity of the nucleus changes, there is a large change in momentum. This causes the nucleus to exert a large force on its surrounding electrons. Because these electrons have a small mass, the consequent change in their momentum corresponds to a large change in their velocity, so they adjust rapidly to the change in nuclear position. The Born-Oppenheimer approximation states that electrons have no momentum and adjust instantaneously to any change in nuclear position. Thus, when the movement of nuclei is considered, the electrons can be assumed to be stationary relative to the nuclei with which they

66 are associated. Likewise, the nucleus can be assumed to be stationary relative to electron

movement, because the nucleus moves much more slowly166(p35).

Further consideration of this approximation depends upon an understanding of the Hamiltonian operator and the Schrödinger equation.

A fundamental particle can be treated like a spherical standing wave with a single peak at the particle’s position. The function which describes this wave is the particle’s wavefunction. The concept of a wavefunction can be extended to a system where the wavefunction, Ψ, describes the state of the system.

In physics and chemistry, an operator is an operation which, when applied to the state of a system, generates a value. When the Hamiltonian operator, ℋ, is applied to the wavefunction of a system, it returns the rate of change of the wavefunction, phase shifted by 90° (multiplied by i, the square root of -1). The shift puts it in phase with the original wavefunction.

ℋ Ψ(𝐫, t) =_2𝜋𝑖ℎ 𝜕

𝜕𝑡Ψ(𝐫, t) 1.17

Where h is Planck’s constant, r is the state of the system and t is time166(p27) .

The Planck-Einstein equation, stated in terms of angular frequency, reveals how the rate of change of a wave is also its energy.

E = ℎ𝜈 = ℎ 2𝜋𝜔 = ℎ 2𝜋 𝑑𝜃 𝑑𝑡𝜃(𝑡) 1.18

Where E is the energy, ν is the frequency (s-1), ω is the angular frequency (rad s-1) and θ is the angular displacement (rad). Consequently, Equation 1.17, the time-dependent Schrödinger equation can be re-written in a time-independent form166(p27):

ℋ Ψ(𝐫) = 𝐸Ψ(𝐫) 1.19

The result of applying the Hamiltonian operator to a system is therefore the total energy of the system and this operator is often equated to the summation of the different energies that contribute to the total energy of the system.

Let the nuclear coordinates be expressed by the vector R and the state of the electrons embodied in vector r. If the Born-Oppenheimer approximation is made, the total wavefunction, Ψ(R, r), can be modelled as the product of two separate nuclear and electronic wavefunctions (Χ and Ψe respectively), calculable using modified Hamiltonian operators, ℋe and ℋn respectively

166(p35) .

Ψ(𝐫, 𝐑) = 𝒳(𝐑)Ψ𝑒(𝐫, 𝐑) 1.20

Schrödinger equations can be formulated for both the electronic (Equation 1.21) and nuclear (Equation 1.22) components166(p27).

67

ℋ𝑒 Ψ𝑒(𝐫) = 𝐸𝑛(𝐑)Ψ𝑒(𝐫) 1.21

ℋ_𝑛 𝒳(𝐑) = 𝐸_𝒳(𝐑)𝒳(𝐑) _1.22

As shown, the coefficients En and EΧ vary with the nuclear positions, R.

ℋ_𝑛= − ℏ

2𝑚∑ ∇2𝑖

𝑁𝑛

𝑖=1

+ 𝒱 1.23

Where ħ is Planck’s constant, h, divided by 2π, µ is the mass of each atom (assumed to be identical in this example), 𝒱 is the interatomic potential, otherwise known as the vibronic energy and  is the del-squared operator or Laplacian, which is here applied to the nuclear wavefunction167(p12). In Euclidean (x, y, z) space, ∇2_≡ 𝜕2 𝜕𝑥2+ 𝜕2 𝜕𝑦2+ 𝜕2 𝜕𝑧2 1.24 𝒱 = E_𝑛− ∑ ∑𝑍𝑎𝑒 𝑟_𝑖𝑎 𝑁𝑛 𝑎=1 𝑁𝑒 𝑖=1 . 1.25

Where Ne is the total number of electrons (over all atoms), Nn is the number of nuclei, Za is the charge of nucleus a, e is the elementary charge (the charge of a proton) and ria is the distance between nucleus a and electron i. En is the electronic energy as shown in Equation 1.21 and which may be calculated using ℋe.

ℋ_𝑒= − ℏ 2𝜇∑ ∇2𝑖(R) 𝑁𝑒 𝑖=1 + ∑ ∑𝑒2 𝑟_𝑖𝑗 𝑖 𝑗=1 𝑁𝑒 𝑖−1 − ∑ ∑𝑍𝑎𝑒2 𝑟_𝑖𝑎 𝑁𝑛 𝑎=1 𝑁𝑒 𝑖=1 + ∑ ∑𝑍𝑎𝑍𝑏𝑒2 𝑟_𝑎𝑏 𝑎 𝑏=1 𝑁𝑛 𝑎=1 1.26

Where μ is the mass of an electron (the reduced mass with respect to the nucleus), rij is the distance between electrons i and j and rab is the distance between nuclei a and b

167(p11)

. Note that the del-squared operator is applied to the electronic wavefunction in this equation, not the nuclear positions R; the value just depends upon R.

In a quantum mechanics simulation, 𝒱 is estimated using the positions of the nuclei and then equation 1.22 is solved, yielding Ex(R) which can be used to work out how the nuclei will move and therefore what 𝒱 will be after a small amount of time has elapsed, the time step.