step toward a geometry of lower energy For additional details about how such geometry optimization searches are performed within modern computational chemistry software see the background Material where this subject was treated in greater detail It often turns out that a molecule has more than one stable structure(isomer) for a given electronic state. Moreover, the geometries that pertain to stable structures of excited electronic state are different than those obtained for the ground state(because the orbital occupancy and thus the nature of the bonding is different). again using arginine an example, its ground electronic state also has the structure shown in Fig. 5. 2 as a stable isomer. Notice that this isomer and that shown earlier have the atoms linked together in identical manners, but in the second structure the a-amino group is involved in two hydrogen bonds while it is involved in only one in the former. In principle, the relative energies of these two geometrical isomers can be determined by solving the electronic Schrodinger equation while placing the constituent nuclei in the locations described in the two figures 6
6 step toward a geometry of lower energy. For additional details about how such geometry optimization searches are performed within modern computational chemistry software, see the Background Material where this subject was treated in greater detail. It often turns out that a molecule has more than one stable structure (isomer) for a given electronic state. Moreover, the geometries that pertain to stable structures of excited electronic state are different than those obtained for the ground state (because the orbital occupancy and thus the nature of the bonding is different). Again using arginine as an example, its ground electronic state also has the structure shown in Fig. 5.2 as a stable isomer. Notice that this isomer and that shown earlier have the atoms linked together in identical manners, but in the second structure the a-amino group is involved in two hydrogen bonds while it is involved in only one in the former. In principle, the relative energies of these two geometrical isomers can be determined by solving the electronic Schrödinger equation while placing the constituent nuclei in the locations described in the two figures
1.916 2 2144 Figure 5.2 Another stable structure for the arginine molecule If the arginine molecule is excited to another electronic state, for example, by promoting a non-bonding electron on its C=O oxygen atom into the neighboring C-O T' orbital. its stable structures will not be the same as in the ground electronic state. In particular, the corresponding C-o distance will be longer than in the ground state, but other internal geometrical parameters may also be modified (albeit probably less so than the C-o distance). Moreover, the chemical reactivity of this excited state of arginine will be different than that of the ground state because the two states have different orbitals available to react with attacking reagents
7 1.916 2.144 Figure 5.2 Another stable structure for the arginine molecule. If the arginine molecule is excited to another electronic state, for example, by promoting a non-bonding electron on its C=O oxygen atom into the neighboring C-O p* orbital, its stable structures will not be the same as in the ground electronic state. In particular, the corresponding C-O distance will be longer than in the ground state, but other internal geometrical parameters may also be modified (albeit probably less so than the C-O distance). Moreover, the chemical reactivity of this excited state of arginine will be different than that of the ground state because the two states have different orbitals available to react with attacking reagents
In summary, by solving the electronic Schrodinger equation at a variety of geometries and searching for geometries where the gradient vanishes and the Hessian matrix has all positive eigenvalues, one can find stable structures of molecules(and ions). The Schrodinger equation is a necessary aspect of this process because the movement of the electrons is governed by this equation rather than by Newtonian classical equations. The information gained after carrying out such a geometry optimization process include(1) all of the interatomic distances and internal angles needed to specify the equilibrium geometry(Rae) and(2)the total electronic energy E at this particular geometry It is also possible to extract much more information from these calculations. For example, by multiplying elements of the Hessian matrix(oE/OR, aR, by the inverse quare roots of the atomic masses of the atoms labeled a and b, one forms the mass- eighted Hessian(m, m)(aE/aR, OR,)whose non-zero eigenvalues give the harmonic vibrational frequencies(o, of the molecule. The eigenvectors (Rka of the mass wieghted Hessian mantrix give the relative displacements in coordinates r a that accompany vibration in the k normal mode (i.e, they describe the normal mode motions). Details about how these harmonic vibrational frequencies and normal modes are obtained were discussed earlier in the Background Material B. Molecular Change-reactions, isomerization, interactions Changes in bonding 8
8 In summary, by solving the electronic Schrödinger equation at a variety of geometries and searching for geometries where the gradient vanishes and the Hessian matrix has all positive eigenvalues, one can find stable structures of molecules (and ions). The Schrödinger equation is a necessary aspect of this process because the movement of the electrons is governed by this equation rather than by Newtonian classical equations. The information gained after carrying out such a geometry optimization process include (1) all of the interatomic distances and internal angles needed to specify the equilibrium geometry {Raeq} and (2) the total electronic energy E at this particular geometry. It is also possible to extract much more information from these calculations. For example, by multiplying elements of the Hessian matrix (¶ 2E/¶Ra¶Rb ) by the inverse square roots of the atomic masses of the atoms labeled a and b, one forms the massweighted Hessian (ma mb ) -1/2 (¶ 2E/¶Ra¶Rb ) whose non-zero eigenvalues give the harmonic vibrational frequencies {wk} of the molecule. The eigenvectors {Rk,a} of the masswieghted Hessian mantrix give the relative displacements in coordinates Rka that accompany vibration in the kth normal mode (i.e., they describe the normal mode motions). Details about how these harmonic vibrational frequencies and normal modes are obtained were discussed earlier in the Background Material. B. Molecular Change- reactions, isomerization, interactions 1. Changes in bonding
Chemistry also deals with transformations of matter including changes that occur when cues react are ex cited(electronically, vibrationally, or rotationally ) or undergo geometrical rearrangements. Again, theory forms the cornerstone that allows experimental probes of chemical change to be connected to the molecular level and that allows simulations of such changes Molecular excitation may or may not involve altering the electronic structure of the molecule. vibrational and rotational excitation do not but electronic excitation ionization, and electron attachment do. as illustrated in Fig. 5. 3 where a bi-molecular reaction is displayed, chemical reactions involve breaking some bonds and forming others and thus involve rearrangement of the electrons among various molecular orbitals Figure 5.3 Two bimolecular reactions a and b show an atom combining with a diatomic c and d show an atom abstracting an atom from a diatomic
9 Chemistry also deals with transformations of matter including changes that occur when molecules react, are excited (electronically, vibrationally, or rotationally), or undergo geometrical rearrangements. Again, theory forms the cornerstone that allows experimental probes of chemical change to be connected to the molecular level and that allows simulations of such changes. Molecular excitation may or may not involve altering the electronic structure of the molecule; vibrational and rotational excitation do not, but electronic excitation, ionization, and electron attachment do. As illustrated in Fig. 5.3 where a bi-molecular reaction is displayed, chemical reactions involve breaking some bonds and forming others, and thus involve rearrangement of the electrons among various molecular orbitals. Figure 5.3 Two bimolecular reactions; a and b show an atom combining with a diatomic; c and d show an atom abstracting an atom from a diatomic
In this example, in part(a) the green atom collides with the brown diatomic molecule and forms the bound triatomic(b). Alternatively, in(c)and(d), a pink atom collides with a green diatomic to break the bond between the two green atoms and form a new bond between the pink and green atoms. Both such reactions are termed bi-molecular because the basic step in which the reaction takes place requires a collision between to independent species(i.e, the atom and the diatomic) A simple example of a unimolecular chemical reaction is offered by the arginine molecule considered above. In the first structure shown for arginine, the carboxylic acid group retains its HOOC-bonding. However, in the zwitterion structure of this same molecule, shown in Fig. 5.4, the HOOC- group has been deprotonated to produce a carboxy late anion group-Coo, with the H ion now bonded to the terminal imine group, thus converting it to an amino group and placing the net positive charge on the adjacer carbon atom The unimolecular tautomerization reaction in which the two forms of arginine are interconverted involves breaking an O-H bond, forming a N-H bond, and changing a carbon-nitrogen double bond into a carbon-nitrogen single bond. In such a process, the electronic structure is significantly altered, and, as a result, the two isomers can display very different chemical reactivities toward other reagents. Notice that, once again, the ultimate structure of the zwitterion tautomer of arganine is determined by the valence preferences of its constitutent atoms as well as by hydrogen bonds formed among various functional groups(the carboxylate group and one amino group and one-NH
10 In this example, in part (a) the green atom collides with the brown diatomic molecule and forms the bound triatomic (b). Alternatively, in (c) and (d), a pink atom collides with a green diatomic to break the bond between the two green atoms and form a new bond between the pink and green atoms. Both such reactions are termed bi-molecular because the basic step in which the reaction takes place requires a collision between to independent species (i.e., the atom and the diatomic). A simple example of a unimolecular chemical reaction is offered by the arginine molecule considered above. In the first structure shown for arginine, the carboxylic acid group retains its HOOC- bonding. However, in the zwitterion structure of this same molecule, shown in Fig. 5.4, the HOOC- group has been deprotonated to produce a carboxylate anion group –COO- , with the H+ ion now bonded to the terminal imine group, thus converting it to an amino group and placing the net positive charge on the adjacent carbon atom. The unimolecular tautomerization reaction in which the two forms of arginine are interconverted involves breaking an O-H bond, forming a N-H bond, and changing a carbon-nitrogen double bond into a carbon-nitrogen single bond. In such a process, the electronic structure is significantly altered, and, as a result, the two isomers can display very different chemical reactivities toward other reagents. Notice that, once again, the ultimate structure of the zwitterion tautomer of arganine is determined by the valence preferences of its constitutent atoms as well as by hydrogen bonds formed among various functional groups (the carboxylate group and one amino group and one –NHgroup)