Molecular mechanics uses an empirically derived system of equations for the potential energy surface, the mathematical form of which is borrowed from classical mechanics. This system of potential functions is called force field. It contains parameters, the numerical value of which is chosen in such a way as to obtain the best agreement between the calculated and experimental characteristics of the molecule. The force field method uses the assumption that it is possible to transfer parameters and force constants characteristic of one molecule to other molecules. In other words, the numerical values ​​of the parameters determined for some simple molecules can be used as fixed values ​​for other related compounds.

Simple molecular mechanics force fields include bond stretching, bond and torsion angle deformation, and van der Waals interactions:

More complex force fields may also include crossover terms, take into account electrostatic interactions, etc.

Molecular mechanics is also often referred to as a computational method using a force field. Force fields were originally developed in spectroscopic research. Later it turned out that they are convenient for use in molecular mechanics. The first such example was central force field, in which only the internuclear distances present in the molecule appear. The cross terms corresponding to the simultaneous change in two internuclear distances are usually neglected, so we get diagonal force field.

Another simple version of the force field is called valence force field. It best corresponds to accepted ideas about the nature of the forces acting in a molecule. The valence force field is specified by the so-called internal coordinates, namely:

All bond lengths;

All independent bond (dihedral) angles:

All independent torsion (azimuthal) angles.

This means that restoring forces act along and across covalent bonds, tending to restore equilibrium bond lengths r, bond angles and torsion angles.

14. Advantages and disadvantages of molecular mechanics methods

The methods of molecular mechanics and molecular dynamics are based on classical concepts. Particles in these cases are considered as material points interacting through so-called force fields, and the force fields themselves are determined by interaction potentials. Molecular mechanics methods use the approach of traditional chemistry. Visually, molecules are represented as a set of balls and rods, with each ball representing an atom, and each rod representing a bond between them. Depending on the type of bonds, interaction potentials are selected, as well as energies and parameters corresponding to certain local configurations of atoms. In this approach to molecular mechanics, potential energy is the sum of terms that describe stretching, bending, and torsion of bonds, as well as electrostatic interactions between unbonded atoms. Allowing an accurate calculation of the geometric structure of molecules and their energy based on available experimental data. It uses the classical idea of ​​chemical bonds between atoms in a molecule and van der Waals forces acting between valence-unbonded atoms.

“-“However, the methods of molecular mechanics can be successfully applied only to a relatively narrow class of molecular structures in configurations close to the equilibrium state.

The methods of molecular mechanics and molecular dynamics are based on classical physics of many particle systems and are not capable of describing quantum effects. Moreover, to obtain numerical results they require detailed knowledge of the interactions between particles, so in each individual case it is necessary to use different models. To obtain realistic results in most cases, additional adjustment of the indicated potentials to experimental data is required. Thus, the ambiguity of the modeling criteria used in molecular mechanics and molecular dynamics limits the widespread application of these methods. At the same time, they make it possible to consider large nanosystems containing up to 10 atoms.

Molecular mechanics is a set of methods for a priori determination of the geometric structure and energy of molecules based on a model in which (unlike the methods of quantum chemistry) the electrons of the system are not explicitly considered. Strictly speaking, Molecular Mechanics (MM) is a method for determining molecular structure based on the representation of a molecule as a set of points of a certain mass held together by classical forces. Molecular mechanics is also often called the method of atom-atomic potential functions. This method is based on the assumption that the energy E of a molecule can be represented by the sum of contributions that can be attributed to bond lengths r, bond angles and dihedral (torsion) angles.

Contributions to molecular energy include elastic binding energy (described by Hooke's law), bending energies of bond angles and spatial angular strains, and energies of electrostatic and van der Waals interactions. In addition, in the general expression for energy there is always a term Eudv, reflecting the van der Waals interaction of valence-unbonded atoms, and a term Ekul, which takes into account the electrostatic interactions of atoms and determines the presence of effective atomic charges.

The molecular mechanics method successfully allows energy minimization for large molecular systems and clusters, at reasonable computational costs. The results of calculations using the molecular mechanics method depend on the parameterization of the force field. The numerical value of the parameters is chosen so as to obtain agreement between the calculated and experimental characteristics of the molecule. The parameters are equilibrium internuclear distances (bond lengths) and bond angles, as well as force constants, that is, the rigidity coefficients of elastic forces connecting pairs of atoms. The simplest models of molecular mechanics take into account bond stretching (Ustr.), deformation of valence (Udef.) and dihedral (torsion) angles (Utors.), interaction of valence-unbonded atoms, also called van der Waals interaction (Uvdv.), electrostatic contributions ( Uel-stat.), etc. :

U = Urast + Udef + Utors + Uvdv + Uel-stat (1)

For each term, a certain analytical expression is written (for example, the energy of the electrostatic contribution Uel-stat., is described by the Coulomb function, but, perhaps, with non-integer charges as parameters) and the parameters of the corresponding functions are adjusted according to some properties of the base molecules. For example, to describe the potential function of saturated hydrocarbons with not very high requirements for calculation accuracy, about ten parameters are sufficient.

The molecular mechanics method can also serve as a computational model for estimating the potential energy of a molecule, taking into account all degrees of freedom. The calculation using the molecular mechanics method consists of minimizing each of the energy contributions, which will give optimal values ​​of distances, bond angles, dihedral angles and energy E of the molecule as a whole. Special computer programs that operate calculations using the molecular mechanics method require much less computer time than quantum chemical calculations, and the accuracy of structure and energy predictions by this method is in most cases comparable to the error of structural and thermochemical measurements.

The method of molecular mechanics allows one to obtain information for a complete description of the geometry of various conformers in the ground state and at saddle points on the potential energy surface (PES), as well as the geometry of the structure in crystals and amorphous (vitreous) bodies. This method is also successfully used to determine the heat of formation, voltage energy, energy of individual conformers and the height of barriers to conformational transformations, to determine vibrational frequencies and dipole moments, to determine electric charge distribution, etc. The range of applications of molecular mechanics is large: from simple molecules to complex metal complexes, polysaccharides and proteins. In combination with other methods, in particular gas electron diffraction and x-ray structural analysis, the reliability and accuracy of determining geometric characteristics increases.

The method of molecular mechanics is not applicable: 1) for modeling systems whose properties are determined by electronic effects such as orbital interactions and 2) in the case of breaking chemical bonds.

Based on calculations by the molecular mechanics method of structural parameters and the energy of molecules in an equilibrium state, it is possible to study the thermodynamic parameters of the systems under study using the Monte Carlo method (a method for solving mathematical problems by modeling random variables and constructing statistical estimates) and the possibility of intra- and intermolecular movements in the system using the molecular dynamics method.

Molecular mechanics is a set of methods for a priori determination of the geometric structure and energy of molecules based on a model in which (unlike the methods of quantum chemistry) the electrons of the system are not explicitly considered. The potential energy surface, which in quantum chemical models is subject to direct calculation, is here approximated by certain empirical functions of varying degrees of complexity, which are, for example, sums of pairwise interaction potentials of atoms. These potential functions, which determine the so-called force field of the molecule, contain some parameters, the numerical value of which is chosen optimally so as to obtain agreement between the calculated and experimental characteristics of the molecule. In the simplest case, the parameters are the equilibrium internuclear distances (bond lengths) and bond angles, as well as force constants, that is, the rigidity coefficients of the elastic forces connecting pairs of atoms. The method is based on the assumption that these parameters can be transferred from one molecule to another, so that the numerical values ​​of the parameters selected for some simple molecules are further used in predicting the properties of other more complex compounds.

The simplest models of molecular mechanics take into account bond stretching, deformation of bond and dihedral (torsion) angles, interaction of valence-unbonded atoms, also called van der Waals interaction, electrostatic contributions, etc.:

U = U rise + U def + U torso + U air + U el-stat.

For each term, a certain analytical expression is written (for example, the energy of the electrostatic contribution U el-stat., described by the Coulomb function, but perhaps with non-integer charges as parameters) and the parameters of the corresponding functions are adjusted according to some properties of the base molecules. For example, to describe the potential function of saturated hydrocarbons with not very high requirements for calculation accuracy, about ten parameters are sufficient.

The sum of all the listed contributions determines the energy U of the molecule as a function of the geometric configuration of the nuclei, and to find the equilibrium geometric configuration of the molecule under study, it is necessary to determine the minimum U using a computer program for searching for stationary points on multidimensional potential surfaces. Thus, the practical actions of the researcher most often come down to setting the starting geometry and calling a program for optimizing geometric parameters from the condition of minimum energy. At the output, the resulting structure is viewed and, if necessary, the energy and its components are analyzed.

It is difficult to overestimate the role of molecular mechanics in modern chemical practice. Since all computational problems relate only to well-developed minimization procedures, even on fairly low-power personal computers it is possible to analyze the structure of large polyatomic molecules in a reasonable time. The ability to see the structure of a molecule on a computer screen, examine it from different angles, check emerging assumptions about steric hindrances, etc. provides invaluable assistance in the work. Molecular mechanics plays the role of a molecular constructor: for a primary assessment of the structure of a molecule of interest to us, it is often easier to assemble a molecule on a computer than to waste time searching for the necessary information in reference literature. When calculating molecular structure at a higher level using quantum chemistry methods, it is useful to use the coordinates of the molecular nuclei found using molecular mechanics as an initial approximation. For many problems, for example, conformational analysis, the level of modeling using molecular mechanics methods turns out to be quite sufficient for qualitative and even quantitative conclusions.

However, the ease of obtaining results sometimes leads to the temptation to use this technique even when it is unacceptable. In each specific case, it is necessary to inquire for which classes of connections the version of the program that is supposed to be used when modeling the properties of a new connection is parameterized. Particular care should be taken in energy estimates, although gross errors are possible for geometric configurations.

The term "molecular mechanics" is currently used to define a widely used method that allows an accurate a priori calculation of the geometric structure of molecules and their energy. This method appeared as a natural continuation of the well-known ideas about chemical bonds between atoms in a molecule and about van der Waals forces acting between valence-unbonded atoms.

According to the Born-Oppenheimer approximation, commonly used in quantum mechanics, the Schrödinger equation for a molecule can be divided into two parts, each of which describes the motion of electrons and nuclei, respectively, and both types of motion can be considered independently of each other. In fact, this is also a good approximation for studying molecules, but it is usually used in a different version. If the electronic structure is being studied, then they do this: they accept certain positions of the nuclei, and then study the electronic structure, considering the positions of the nuclei unchanged. In molecular mechanics, they use the opposite approach: they study the motion of nuclei, but electrons are not explicitly considered at all; it is simply assumed that they are optimally distributed in space around the nuclei.

Based on the Born-Oppenheimer approximation, molecular mechanics (it is often called classical, although it borrowed only potential functions from classical mechanics) can be placed on a quantum mechanical basis in relation to the nuclei of atoms, since electrons are not explicitly considered, but are considered only the cause of the emergence of that potential field , in which the nuclei are located. The potential itself is determined empirically. Further, the results of calculations of the geometric structure and energy, as well as vibration frequencies in the harmonic approximation, are assumed to be the same regardless of the use of the quantum mechanical or classical approach.

To implement the quantum mechanical approach to studying the structure of molecules, it is necessary to make a number of simplifications. For this purpose, the self-consistent Hartree-Fock field approximation is most often used. The accuracy of first-principles calculations for the geometric structure of molecules varies over a fairly wide range (from moderate to very high) depending on the type of atomic wave functions used. Thus, if calculations are carried out with an extended basis set, including ((-orbitals for elements of the second period and p-orbitals of the hydrogen atom, then structural and energy data are obtained with an accuracy comparable to the results of molecular mechanics, in which the limit of accuracy is not determined by the calculation method, but by the difference in the strict physical definition of the properties of molecules, for example, the difference in bond lengths.It should be noted that detailed quantum mechanical calculations of the geometric structure of molecules can be performed for any molecule or any fragment without involving any experimental information about the system being studied.

As for molecular mechanics, there are a large number of parameters used in the calculations, which for any given molecule must be known from previous studies of other molecules of the same class. Thus, the scope of molecular mechanics is limited in the sense that the molecule being studied must belong to a previously studied class of compounds. For quantum mechanical calculations “from first principles” there are no such restrictions, which makes them especially attractive for studying truly new types of molecules.

The energy of a molecule in the ground electronic state is a function of its nuclear coordinates, which can be represented by a potential energy surface. In the general case, there may be energy minima in the potential surface, separated by low maxima.

When considering structures located on a potential surface, it is useful to adhere to certain terminology. Each point related to the energy minimum corresponds to a conformer. To move from one minimum to another, the molecule must pass through the saddle point (pass) separating them. At the saddle point, a shaded (obscured) conformation with slightly stretched bonds and noticeably deformed bond angles is realized. Movement along the molecular coordinate, which links different conformations, from the point of view of internal coordinates corresponds to rotation about the central bond. The energy of the system increases as it approaches a saddle point, reaches a maximum, and then decreases as it approaches another minimum.

Complex molecules in general can have many energy minima varying in depth. To a first approximation, such a molecule is characterized by a structure corresponding to the deepest energy minimum. The next approximation is to describe an equilibrium mixture of molecules located at all energy minima in accordance with the Boltzmann distribution. Another, more accurate approximation no longer considers the molecules motionless, located at points with minimum energy, but takes into account their vibrational motion along some part of the surface near the minimum energy. Finally, thermal motion can transport some molecules through saddle points from one minimum to another at a speed corresponding to the Gibbs free energy of activation.

If a molecule consisting of N atoms and described by ZgU coordinates x 1U deforms with respect to its equilibrium configuration with energy?/ 0 and coordinates x 0, then its potential energy can be represented by a Taylor series expansion:

The potential energy of a molecule is entirely electromagnetic in nature and is usually given as a sum of individual components:

which correspond to the following types of interactions: and b- potential energy of valence bonds; С/„ - bond angles; S/f - torsion angles; and g- flat groups; and pu- van der Waals forces; V e1- electrostatic forces; and no- hydrogen bonds. These components have different functional forms.

Valence bonds are maintained by potential

where r is the bond number in the molecule; LG 6 - total number of valence bonds; K b1- effective rigidity of the valence bond; g 1- connection length; g 0 - equilibrium bond length.

Comparison of parabolic (i) and real (2) valence bond potentials

Replacing the real potential describing valence interactions with a parabolic one (Fig. 5.4) is justified by the fact that at room temperatures the vibrations of valence bonds are small. At the same time, in a number of problems it is necessary to carry out model calculations at high temperatures, and then the use of the parabolic potential does not lead to the breaking of valence bonds.

Bond angles are given by the potential

Where I- bond angle number; N^- total number of bond angles; TO,. I- effective elasticity of the bond angle; a, is the value of the bond angle; and 0 is its equilibrium value.

The energy of torsion interactions and potentials corresponding to flat groups is written in the same form:

where cf is the number of the torsion angle; I- harmonic number; K 0; - constant; # f g - harmonic contribution to the torsion angle potential; l f, - harmonic multiplicity. Potentials?/f i And[ differ by constants.

Van der Waals interactions of atoms separated by three or more valence bonds are described by Lennard-Jones potentials:

Potential parameters AiV depend on the types of atoms z and y participating in the interaction; g, = |g, - - g y |, where g, and g y are the coordinates of interacting atoms.

Electrostatic interactions are specified by the Coulomb potential

Where, q j- partial charges on atoms; p is the dielectric constant of the medium.

Hydrogen bonds appear and disappear during the movement of atoms between those that have electrostatic interactions. The functional form of the hydrogen bond potential is similar to the potential of van der Waals interactions (Fig. 5.5), but with shorter-range attractive forces:

The hydrogen bond is a special type of bond and is due to the fact that the radius of the H + ion is an order of magnitude smaller than that of other ions. In formulas (5.39) and (5.41) there is a difference in the contributions describing the attraction. Addiction

Comparison of potentials for hydrogen bonding and van der Waals interaction

V/g*; in (5.39) corresponds to the dispersive dipole-dipole interaction, and the quantity IN'/ g (F in (5.41) is introduced based on experimental data.

Note that the system of potentials (5.36)-(5.41) is a very approximate way of specifying potential energy. Its disadvantages are that the interaction energy is represented as a sum of paired spherically symmetric interactions. Both of these are, generally speaking, incorrect, but this has to be assumed for now due to the absence of other dependencies.

Methods for searching for the equilibrium structure of a molecule for which the condition that the first derivatives of the potential energy are equal to zero is simultaneously satisfied And in all coordinates x, (di/dx 1 = 0), can be divided into two groups: minimization with first derivatives (linear methods) and with second derivatives (quadratic methods). Simple search methods consider only the slope of the potential surface (i.e., the first derivative), calculated numerically or analytically, while more complex minimization methods use both the slope and the curvature of the potential surface (i.e., the first and second derivatives).

Varying the geometry changes the energies of different conformations by different amounts, which has consequences for conformational equilibria, especially for strained molecules. The largest energy changes with varying geometry are usually observed for transition states in the process of conformational transformations (interconversions). Internal rotation barriers, for example, are often too high, sometimes even by several times, if a rigid rotator approximation is used.

Several remarks should be made regarding the geometric model that is determined by any minimization procedure. This procedure is an iterative geometry optimization. Therefore, if there are several potential wells for a molecule, the found minimum energy will depend on the initial approximation, i.e., on the potential well to which the rough initial structure belongs. There are no general methods for finding global energy minima, and those minima that are determined are usually local. Therefore, as initial approximations, a number of reasonable possible conformations are identified, such as armchair, bathtub, and twisted bathtub, but for large molecules the number of test structures can be quite large. Thus, until a systematic study of all geometrically possible structures has been carried out, the minimum energy conformation found may depend on the subjective choice of the starting structure by a particular researcher.

The geometric structure of a molecule can be determined in several ways. The answer to the question which one is better depends on whether it is necessary to know the geometry of the molecule in a hypothetical state, when it is completely motionless (this is the state of the molecule at the bottom of the potential well that is obtained when optimizing the geometry using methods “from first principles”), or whether there is interest in finding observable quantities similar to those measured by diffraction and spectroscopic methods. Differences in determination methods and corresponding differences in geometric structure are due to the vibrational movement of molecules. A similar problem exists for molecular energy, so it is necessary to understand the meaning of steric energies derived from molecular mechanics calculations and their relationship to observed energy characteristics.

Molecular mechanics uses an empirically derived system of equations for the potential energy surface, the mathematical form of which is borrowed from classical mechanics. This system of potential functions, called force field, contains some variable parameters, the numerical value of which is chosen in such a way as to obtain the best agreement between the calculated and experimental characteristics of the molecule, such as geometric structure, conformational energies, heats of formation, etc. The force field method uses one general assumption about the possibility of transferring the corresponding parameters and force constants characteristic of one molecule on other molecules. In other words, the numerical values ​​of the parameters determined for some simple molecules are then used as fixed values ​​for the remaining related compounds.

Simple force fields of molecular mechanics simultaneously include bond stretching, bond and torsion angle deformation, and van der Waals interactions:

The summation covers bond and torsion (dihedral) angles, as well as all interactions between valence-unbonded atoms, except the interactions of atoms bonded to the same common atom.

More complex force fields can include, in addition to the connections indicated in equation (5.42), also cross terms, take into account electrostatic interactions, etc. For each of these cases there is a first approximation, and in many cases higher approximations have been developed. The sum of all these terms is called steric energy of the molecule.

Since there is a close relationship between the structure of a molecule and its energy, molecular mechanics always considers them together. Indeed, in order to obtain the structure, it is necessary to study the energy of the system and find the minimum of this function. Even if only the structures of molecules and their relative energies, i.e., conformations and conformational energies, are calculated, it can be considered that chemically important information has been obtained.

The vibrational motion of a diatomic molecule, or any pair of atoms bonded together, is often described using the Morse function. Its shape resembles a parabola near the minimum, where Hooke's law is satisfied, but at short distances the energy increases more rapidly, and at long distances - more slowly. It follows that with increasing temperature, when excited vibrational levels are populated, the bond lengths will increase.

It should be noted that the definition of “link length” has different interpretations. In particular, internuclear distances are obtained from electron diffraction measurements, usually denoted as g a. Designation g e used for the internuclear distance in a rigid model, when each of the atoms in the molecule is at the bottom of its potential well. This value is found directly from quantum mechanical calculations "from first principles", but this distance is difficult to determine experimentally.

Molecular mechanics is also often referred to as a computational method using a force field. Force FIELDS, which were originally developed by spectroscopists in a more physically rigorous form, then began to be used in molecular mechanics. The first such example was the central force field, in which only the internuclear distances present in the molecule appear. Cross terms corresponding to the simultaneous change of two internuclear distances are usually neglected, so a diagonal force field is obtained. Physically, this corresponds to a model in which harmonic forces act between all possible pairs of atoms, regardless of whether they are connected by a chemical bond or not. However, this approach, which makes sense for ionic crystals and not for molecules of organic compounds, has not found wide application.

Another simple version of the force field, which best corresponds to accepted ideas about the nature of the forces acting in a molecule, is called valence force field. It is specified by internal coordinates, which are usually all bond lengths and a set of independent bond and torsion (dihedral or azimuthal) angles. This means that restoring forces act along and across covalent bonds, trying to restore equilibrium bond lengths r, bond angles A and torsion angles cf. With this choice of coordinates, the simplest approximation, although very rough, is to neglect all non-diagonal force constants. The result is a force field consisting of harmonic potentials according to Hooke's law:

where / are functions describing the components of the force field.

According to the basic model used in molecular mechanics, the atoms in a molecule are, as it were, bound together by separate independent springs that retain the “natural” values ​​of bond lengths and bond angles. Then, as in the case of a diagonal force field, one can use (according to Hooke’s law) the harmonic potential functions represented by equation (5.44) for bond stretching and (5.45) for bond angle deformations:

Any force field of molecular mechanics contains these functions. For large deformations, deviation from the harmonic approximation is to be expected, in which case the Morse function is an example of a more general potential. However, the Morse function is not usually used in molecular mechanics due to the excessive consumption of computer time. Therefore, simpler means can be recommended that nevertheless give results of the same quality. Theoretically, the most attractive technique is to terminate the Taylor series represented by Equation (5.34) after the term following the quadratic relationships for bond stretching and bond angle strains. A potential function with a cubic term as in Eq.

has acceptable properties in a certain range and is well applicable to abnormally long bonds.

To account for nonbonded interactions, vibrational spectroscopy and molecular mechanics usually use potential functions that are very different from each other. Molecular mechanics allows for the possibility of transferring force constants to systems with strong non-bonded interactions. This procedure is based on the assumption of equality of intra- and intermolecular non-valence interactions, therefore using potential functions originally derived for noble gas atoms and usually known as van der Waals interactions. Van der Waals' model of a real gas considers atoms as impenetrable "hard spheres". However, the nonbonded interaction potentials discussed below correspond to "soft sphere" potentials, thereby improving molecular mechanics over the "hard sphere" approximation.

As is known, the general form of any van der Waals potential function consists of two components: the repulsive force at a short distance and the attractive force at a long distance, which asymptotically tend to zero at very large distances. The main characteristics of such a function (see Fig. 5.4) are:

• ? the distance at which the energy is minimal, which is related to the van der Waals radius;
• ? the depth of the potential well, which is due to polarizability;
• ? the steepness of the left side of the curve corresponding to repulsion, which is associated with the rigidity of the potential.

As already noted, atoms of noble gases and non-polar molecules with closed electron shells interact over large distances using induced electrical moments - dispersion forces. Second-order perturbation theory gives a potential corresponding to this attraction and described by the equation

The first term, representing the interaction energy between the instantaneous and induced dipoles, dominates in magnitude at large distances, so the attractive energy is usually described by only this one term (r -6). The coefficient c 6 is chosen to take into account the neglect of higher members. If a molecule has a permanent charge or dipole moment, then the interaction terms corresponding to permanent moments are definitely larger than those corresponding to induced moments.

In classical electrostatic theory, the interaction energy between diffuse charged clouds is calculated by expanding into another series containing charge-charge, charge-dipole, dipole-dipole, and higher terms interactions.

Coulomb interaction described by the equation

is the dominant type of interaction between ions. For uncharged polar molecules, the energy of the dipole-dipole interaction is the main term, which, according to classical electrostatics, can be calculated using the so-called Jeans formula:

Electrostatic dipole-dipole interaction

where?) is the effective dielectric constant;

X- the angle between two dipoles p, and p y -; a, and ay are the angles formed by the dipoles with the vector that connects them, as shown in Fig. 5.6.

Many attempts have been made to construct a force field of molecular mechanics only on the basis of bond stretching potentials, angular deformations and van der Waals interactions, varying the parameters and the type of potential functions. However, it turned out to be impossible to obtain even approximately the correct value of the energy difference between the staggered (inhibited) and shaded conformations in some molecules using van der Waals parameters taken from the scattering of molecular beams or from the results of studies of crystal packing. If these parameters were chosen to reproduce the internal rotation barrier, then they gave unacceptable results when calculating other properties.

Completely different solutions were proposed, but the most fruitful idea was to introduce, within the framework of molecular mechanics, based on van der Waals interactions, a corrective function to describe the internal rotation of the molecule relative to simple bonds - the torsion potential in the form of the following equation:

Of course, equation (5.50) is only a first approximation for describing torsion energy, which acts as a corrective term in the force field. In this cosine function, the minima correspond to staggered conformations and the maxima correspond to shaded conformations. It is believed that torsional energy arises due to repulsion between bonds, which is not taken into account by van der Waals interactions. An alternative, but not entirely satisfactory, explanation is the idea of ​​torsional energy as a correction for the anisotropy of van der Waals repulsive forces, which is more pronounced at small than at large angles to the chemical bond. As a result of numerous quantum chemical works, other explanations have been proposed.

The heat of formation is a fundamental energy characteristic of a molecule. Experimental studies of heats of formation, together with attempts to predict their values, have made a significant contribution to the theory of structural chemistry. In molecular mechanics, it is believed that the heat of formation consists of the following components: the formation of chemical bonds; stress effects represented by steric energies; statistical-thermodynamic factor due to the population of vibrational levels; mixing of conformers and other reasons.

Using statistical thermodynamics, it is possible to calculate the thermally averaged enthalpy of one conformer of a certain molecule, counting it from the bottom of the potential well. If there are several conformers, then the enthalpy of the mixture (/T cm) is found from the mole fractions (./V,) and heat (N:) existing conformers according to the equation

Mole fractions are obtained from the Boltzmann distribution

Where g i is the statistical weight of the conformer g, i.e., the number of identical conformations; And C* is the Gibbs free energy without entropy effects due to the presence of identical conformations.

In order to find the heat of formation of a given compound, which is a common characteristic for comparisons, it is necessary to know the absolute energy corresponding to the bottom of the potential well. In principle, it can be found by summing the binding energies (BE) and steric energy (SE) of the molecule, calculated using a force field. Bond energies are determined empirically by reproducing the known heats of formation of simple compounds.

Of all the thermodynamic functions, not only enthalpy can be found using molecular mechanics. Knowing the geometric structure, atomic masses and vibration frequencies, the method of statistical thermodynamics can calculate entropy, Gibbs energies and the equilibrium constant. To calculate entropy, it is necessary to calculate the total energy for the following types of motion using standard methods of statistical thermodynamics: translational (depending on the masses of atoms), rotational (depending on the moments of inertia determined by the geometry of the molecule) and vibrational (depending on the vibration frequencies).

Heats of formation are very useful in comparing the relative energies of isomers. However, when comparing compounds that are not isomers, the values ​​of steric energy and heat of formation are not useful. Therefore, to make quantitative comparisons in such cases, a characteristic called tension energy.

Quantitative theories of stress are based on the application of the bond energy diagram. The basic idea is that there are some simple molecules that lack tension. Large molecules are stress-free only if their heats of formation can be represented as the sum of bond energies and other indicators calculated from the “unstressed” small molecules. If the actual heat of formation is greater than that predicted based on the calculation of these “unstressed” systems, then such a molecule is considered to be stressed. Thus, using molecular mechanics, it is possible to calculate the stress energy, since the corresponding strain energies occurring in the molecules are calculated using the energy minimization procedure.

In some cases, it is convenient to introduce a new quantity, the so-called intrinsic stress, which is preliminary in nature in the process of finding the total stress energy. The intrinsic voltage is equal to the sum of the bond parameters and the heat of formation of the most stable conformer, which are calculated under the following two assumptions: 1) the compound exists as a single conformer and 2) the molecule does not contain open-chain C-C bonds that require the introduction of special torsion corrections. Thus, the total voltage is the sum of the own voltage and two additional corrections. Self-tension is of interest for some purposes, and total tension for others. By defining these two concepts separately, all available information can be separated, making it easier to analyze complex systems.

One of the ideas closely related to the concept of tension relates to the explanation and prediction of molecular stability, but often stability does not follow directly from tension. Highly strained molecules are sometimes very stable due to high activation energies for those reactions that lead to less strained compounds.

Molecular mechanics was originally intended to calculate the geometric structure and energy state of isolated molecules, that is, molecules in the gas phase. However, it is not surprising that already at an early stage of development of the method, which takes into account the importance of non-bonded interactions, attempts were made to apply these calculations to the determination of crystal packing and the structure of solids. It has been suggested that the same potential functions and their parameters also apply to intra- and intermolecular nonvalent interactions.

The field of application of calculations related to the structure of crystals is extremely wide and is now in the stage of active development. It extends from studying the influence of crystal packing on the structure of molecules to determining the thermodynamic and dynamic properties of crystals. For the method of molecular mechanics itself, these calculations of crystal packings are of particular importance in determining the parameters of non-bonded interactions.

Geometry optimization in a crystal can be accomplished in three different ways.

The first is to preserve the unit cell parameters, for example by placing molecules in locations found experimentally, and only perform optimization within the molecule. Such calculations are intended to clarify the influence of crystalline forces on the structure of molecules.

In the second method, the intramolecular coordinates are kept unchanged and the six parameters of the unit cell are varied (three sides and three angles). However, such calculations of crystal packing contain significant uncertainty due to the fact that the molecules are considered rigid entities.

The third and most powerful method is to simultaneously optimize intra- and intermolecular coordinates.

Molecular structures obtained from gas measurements using microwave spectroscopy or electron diffraction usually agree very well with what is found for crystals. The deviations that arise in this case, as well as when comparing the structure of molecules in crystals with the calculated and theoretical results of quantum chemistry and molecular mechanics, are most often explained by the effects of crystal packing. In some cases, direct experimental verification of the packing effect is possible if there is more than one independent molecule in the unit cell or if the molecule forms two different types of crystals (polymorphism). Then the differences in the structures of such molecules are associated with the influence of packing forces.

The theoretical consideration begins with the basic proposition: a molecule in a crystal must obtain such a structure as to have a minimum energy in the force field of its surrounding neighbors. To do this, certain positions of molecules with approximately correct geometry are specified in the crystal and energy is minimized depending on the intramolecular coordinates.

The dimensions of the unit cell of a crystal are determined by intermolecular nonvalent interactions. Therefore, potential molecular mechanics functions should be suitable for calculating the unit cell size. For this reason, crystal properties are important experimental criteria in determining potential functions.

Van der Waals interactions between two molecules in a crystal are relatively small compared to the many similar intramolecular interactions encountered in molecular mechanics. As the molecules move away from each other, these small interactions decrease in proportion to 1/g 6. This leads to the idea that when studying a crystal, it can be represented as a cluster of blocks containing only a few nearest neighbors.

MOLECULAR MECHANICS

(method of atomic potentials), calculated empirical. method for determining geom. characteristics and energy of molecules. Based on the assumption that the energy of the E molecule may. represented by the sum of deposits, which may be are related to bond lengths r, bond angles a and dihedral (torsion) angles t (the corresponding energy components are denoted E st, E shaft and E torus). In addition, in the general expression for energy there is always a term E vdv, reflecting the van der Waals interaction. valence-unbonded atoms, and the E cool member , taking into account electrostatic interaction atoms and determining the presence of effective atomic charges. Thus, the total energy of the molecule is represented by the sum:

To calculate the first two terms, the well-known from mechanics is most often used Hooke's law(hence the name of the method):

Analyst. expression for the energy E torus, for example. for a C 2 H 6 molecule, has the form:

Where potential barrier internal rotation. The energies E vdv and E cool are calculated using the Lennard-Jones or Buckingham formulas for model potentials (see. Intermolecular interactions, Nonvalent interactions). Options a, r 0, a 0, etc. in all equations used are selected in such a way as to satisfy the experiment. structural and thermochemical. data for the simplest molecules chosen as standards (for hydrocarbons, the standard molecules are CH 4, C 2 H 6 and certain others). The resulting set of parameters is then used to calculate the characteristics of molecules of a certain class of compounds. (for example, saturated hydrocarbons, alcohols, etc.), as well as for the study of unstudied substances. Calculation using the M. m. method consists of minimizing each of the energy. deposits, which gives optimal. values ​​of r, a and t and the energy of the E molecule as a whole. Specialist. Computer programs require much less computer time than quantum chemical ones. calculations, and the accuracy of predictions is comparable to the error of structural and thermochemical. measurements.

The M. m. method allows one to obtain information for a complete description of the geometry of decomposition. conformers in the main condition and at saddle points on the surface of the potential. energy (PPE), as well as geom. structures in a crystal. The heats of formation, tension energies, energies of individual conformers, and the heights of barriers to conformations are also determined. transformations, oscillation frequency, electrical distribution. charge, dipole moments, chemical shifts in NMR spectra, chemical rates. r-tions, etc. The range of application of M. m. is large: from simple molecules to polysaccharides and proteins. In combination with other methods, in particular gas electron diffraction and x-ray structural analysis, the reliability and accuracy of geom determination. characteristics increases.

Based on the calculation of the structural parameters and energy of molecules in an equilibrium state, the possibilities of intra- and intermolecular movements that form the subject of study they say. speakers.

M. m. was developed in the 60s. 20th century T. Hill and A. I. Kitaigorodsky. The term was proposed by L. Bartell in 1958.

Lit.: Dashevsky V.G., Conformational analysis of organic molecules, M., 1982; Burkert U., Ellinger N. L., Molecular mechanics, trans. from English, M., 1986; KoffiU., Ivens M., Grigolini P., Molecular and Spectra, trans. from English, M., 1987. B.C. Mastryukov.

Chemical encyclopedia. - M.: Soviet Encyclopedia. Ed. I. L. Knunyants. 1988 .

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Books

• Mechanics and molecular physics in the course of general physics. Textbook on general physics, Landau L.D.. The book is a concise introduction to the most important fundamental concepts and models in mechanics and molecular physics. It will no doubt be useful as an introductory textbook for...