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676 12. The Molecule in an Electric or Magnetic Field MAGNETIC PHENOMENA • An elementary particle has a magnetic dipole moment M proportional to its spin angular momentum I,i.e.M = γI,whereγ stands for what is called the gyromagnetic factor (characteristic for the kind of particle). • The magnetic dipole of a particle with spin I (corresponding to spin quantum number I) in homogeneous magnetic field H has 2I + 1 energy states E m I =−γm I ¯ hH,where m I =−I −I +1+I. Thus, the energy is proportional to H. • The Hamiltonian of a system in an electromagnetic field has the form ˆ H = j=1 1 2m j ˆ p j − q i c A j 2 +q j φ j + ˆ V where A j and φ j denote the vector and scalar fields at particle j (both are functions of position in the 3D space) that characterize the external electromagnetic field. • A and φ potentials contain, in principle (see Appendix G), the same information as the magnetic and electric field H and E. There is an arbitrariness in the choice of A and φ. • In order to calculate the energy states of a system of nuclei (detectable in NMR spec- troscopy) we have to use the Hamiltonian ˆ H given above, supplemented by the inter- action of all magnetic moments related to the orbital and spin of the electrons and the nuclei. • The refinement is based on classical electrodynamics and the usual quantum mechanical rules for forming operators (Chapter 1) or, alternatively, on the relativistic Breit Hamil- tonian (p. 131). This is how we get the Hamiltonian (12.66) which contains the usual non-relativistic Hamiltonian (12.67) plus the perturbation (12.68) with a number of terms (p. 657). • Experimentalists use an empirical Hamiltonian (eq. (12.82)), in which they have the interaction of the nuclear spin magnetic moments with the magnetic field (the Zee- man effect), the latter weakened by the shielding of the nuclei by the electrons plus the dot products of the nuclear magnetic moments weighted by the coupling con- stants. The experiment gives both the shielding (σ A ) and the coupling (J AB )con- stants. • Nuclear spin coupling takes place through the induction mechanism in the chemical bond (cf. Figs. 12.13, 12.15). Of key importance for this induction is high electron density at the position of the nuclei (the so called Fermi contact term, Fig. 12.13). • The theory of shielding and coupling constants was given by Ramsey. According to the theory, each quantity consists of diamagnetic and paramagnetic contributions. The diamagnetic term is easy to calculate, the paramagnetic one is more demand- ing. • Each of the contributions to the shielding constant individually depends on the choice of the origin of the vector potential A, while their sum is invariant with respect to this choice. • The London atomic orbitals χ L = exp(−iA C · r)χ(r − R C ) used in calculations for a molecule in a magnetic field depend explicitly on that field, through the value A C of the vector potential A calculated at the centre R C of the usual atomic orbital χ(r −R C ). • The most important feature of London orbitals is that all the integrals appearing in calcu- lations are invariant with respect to the origin of the vector potential. This is why results obtained using London orbitals are also independent of that choice. Main concepts, new terms 677 Main concepts, new terms Hellmann–Feynman theorem (p. 618) Cartesian multipole moments (p. 624) dipole, quadrupole, octupole moments (p. 624) multipole polarizability (p. 628) multipole hyperpolarizability (p. 628) induced dipole moment (p. 628) ZDO (p. 635) sum over states method (p. 635) finite field method (p. 639) Sadlej relation (p. 640) second/third harmonic generation (p. 646) nuclear magnetic dipole (p. 648) spin magnetic moment (p. 648) gyromagnetic factor (p. 648) Bohr magneton (p. 648) nuclear magneton (p. 648) Maxwell equations (p. 962) NMR (p. 658) NMR Hamiltonian (p. 658) shielding constants (p. 659) spin–spin intermediate coupling (p. 659) local field (p. 659) chemical shift (p. 664) Ramsey theory (p. 666) diamagnetic effect (p. 668) paramagnetic effect (p. 668) coupling constant (p. 668) direct spin–spin interaction (p. 669) diamagnetic spin–orbit contribution (p. 669) paramagnetic spin–orbit (p. 670) spin–dipole contribution (p. 670) Fermi contact contribution (p. 670) coupling mechanism (p. 672) London orbitals (p. 673) GIAO (p. 673) From the research front The electric dipole (hyper)polarizabilities are not easy to calculate, because: • the sum over states method (SOS) converges slowly, i.e. a huge number of states have to be taken into account, including those belonging to a continuum; • the finite field method requires a large quantity of atomic orbitals with small exponents (they describe the lion’s share of the electron cloud deformation), although, being diffuse, they do not contribute much to the minimized energy (and lowering the energy is the only indicator that tells us whether a particular function is important or not). More and more often in their experiments chemists investigate large molecules. Such large objects cannot be described by “global” polarizabilities and hyperpolarizabilities (ex- cept perhaps optical properties, where the wave length is often much larger than size of molecule). How such large molecules function (interacting with other molecules) depends first of all on their local properties. We have to replace such characteristics by new ones offering atomic resolution, similar to those proposed in the techniques of Stone or Sokalski (p. 1018), where individual atoms are characterized by their multipole moments, polariz- abilities, etc. Even a few years ago, the shielding and especially spin–spin coupling constants were very hard to calculate with reasonable accuracy. Nowadays these quantities are computed routinely using commercial software with atomic London orbitals (or other than GIAO basis sets). The current possibilities of the theory in predicting the nuclear shielding constants and the nuclear spin–spin coupling constants are shown in Tables 12.1 and 12.2. Note that the accuracy of the theoretical results for shielding constants is nearly the same as that of ex- periment. As to the spin–spin coupling constants, the theoretical results are only slightly off experimental values. 678 12. The Molecule in an Electric or Magnetic Field Table 12.1. Comparison of theoretical and experimental shielding constants. The shielding constant σ A (unitless quantity) is (as usual) expressed in ppm, i.e. the number given has to be multiplied by 10 −6 to obtain σ A of eq. (12.83). The Hartree–Fock, MP2, MP4 results are calculated in J. Gauss, Chem. Phys. Letters 229 (1994) 198; the CCSD(T) in J. Gauss, J.F. Stanton, J. Chem. Phys. 104 (1996) 2574, and the CASSCF in K. Ruud, T. Helgaker, R. Kobayashi, P. Jørgensen, K.L. Bak, H.J. Jensen, J. Chem. Phys. 100 (1994) 8178. For the Hartree–Fock method see Chapter 8, for the other methods mentioned here, see Chapter 10. The references to the corresponding experimental papers are given in T.Helgaker,M.Jaszu ´ nski, K. Ruud, Chem. Rev. 99 (1999) 293. The experimental error is estimated for σ H in ammonia as ±10, for σ O as ±172, for σ H in water as ±0015, for σ F as ±6, for σ H in hydrogen fluoride as ±02 CH 4 NH 3 H 2 OHF Method σ C σ H σ N σ H σ O σ H σ F σ H Hartree–Fock 1948317 2623317 3281307 4136284 MP2 2010314 2765314 3461307 4242289 MP4 1986315 2699316 3375309 4187291 CCSD(T) 1989316 2707316 3379309 4186292 CASSCF 20043119 26963102 33533021 41962849 experiment 198.7 30.61 264.54 31.2 344.0 30.052 410 28.5 Table 12.2. Comparison of theoretical and experimental spin–spin coupling constants n J AB for eth- ylene (n denotes the number of separating bonds), in Hz. For the methods used see Chapter 10. All references to experimental and theoretical results are in T. Helgaker, M. Jaszu ´ nski, K. Ruud, Chem. Rev. 99 (1999) 293 Spin–spin coupling constants J AB for ethylene, in Hz Method 1 J CC 1 J CH 2 J CH 2 J HH 3 J HH-cis 3 J HH-trans MC SCF 719 1466 −30 −27109181 EOM-CCSD 701 15323 −295 044 1157 1780 experiment 67457 156302 −2403 2394 11657 19015 Ad futurum. . . It seems that the SOS method will be gradually sent out of business. The finite field method (in the electric field responses) will become more and more important, due to its simplicity. It remains however to solve the problem, how to process the information we get from such computations and translate it into the above mentioned local characteristics of the molecule. Contemporary numerical methods allow routine calculation of polarizability. It is diffi- cult with the hyperpolarizabilities that are much more sensitive to the quality of the atomic basis set used. The hyperpolarizabilities relate to non-linear properties, which are in high demand in new materials for technological applications. Such problems as the dependence of the molecular spectra and of the molecular con- formations and structure on the external electric field (created by our equipment or by a neighbouring molecule) will become more and more important. The theory of the molecular response to an electric field and the theory of the molecular response to a magnetic field, despite some similarities, look as if they were “from another story”. One of the reasons is that the electric field response can be described by solving the Schrödinger equation, while that corresponding to the magnetic field is based inherently Additional literature 679 on relativistic effects. The latter ones are much less investigated except some quite simple examples. Another reason may be the scale difference: the electric effects are much larger than the magnetic ones. However, the theory for the interaction of matter with the electromagnetic field has to be coherent. The finite field method, so gloriously successful in electric field effects, is in the “stone age” stage for magnetic field effects. The propagator methods 84 look the most promising, these allow for easier calculation of NMR parameters than the sum-over-states methods. Additional literature A.D. Buckingham, Advan. Chem. Phys. 12 (1967) 107. A classical paper on molecules in a static or periodic electric field. H.F. Hameka, “Advanced Quantum Chemistry. Theory of Interactions between Mole- cules and Electromagnetic Fields”, Addison-Wesley Publishing Co., Reading, Massa- chusetts, USA (1965). This is a first class book, although it presents the state of the art before the ab initio methods for calculating the magnetic properties of molecules. T.Helgaker,M.Jaszu ´ nski, K. Ruud, Chem. Rev. 99 (1999) 293. A competent review article on the magnetic properties of molecules (NMR) with pre- sentation of suitable contemporary theoretical methods. Questions 1. The Hellmann–Feynman theorem says that ( ˆ H means the Hamiltonian depending on the parameter P): a) ∂E ∂P =ψ| ∂ ˆ H ∂P |ψ,ifψ is the variational trial function; b) ∂E ∂P =ψ| ∂ ˆ H ∂P |ψ,ifψ is the Hartree–Fock wave function; c) ∂E ∂P =ψ| ∂ ˆ H ∂P |ψ, for any normalized ψ; d) ∂E ∂P =ψ| ∂ ˆ H ∂P |ψ,ifψ is an eigenfunction of ˆ H. 2. The proportionality constant at the third-power term (the powers of electric field in- tensity) in the expression for the energy of a molecule in a homogeneous electric field is: a) a component of the quadrupole moment; b) a component of the dipole moment; c) − 1 3! β,whereβ denotes a hyperpolarizability component; d) a component of the octupole moment. 3. A non-polar molecule (with a non-zero quadrupolar moment) in an electric field with a non-zero gradient: a) does not interact with the field; b) will rotate to align its dipole moment along the field; c) will orient to align its longer quadrupole axis along the field; d) will orient to align its longer quadrupole axis along the gradient of the field. 84 J. Linderberg, Y. Öhrn, “Propagators in Quantum Chemistry”, 2nd edition, John Wiley & Sons, Ltd, 2004. 680 12. The Molecule in an Electric or Magnetic Field 4. Second harmonic generation requires that the molecule has: a) large dipole hyperpolarizabilities; b) large quadrupole and octupole polarizabilities; c) large octupole and higher order hyperpolarizabilities; d) a large quadrupole hyperpolarizability. 5. In variational calculations for the benzene molecule (zero electric field) the GTO expo- nents and positions have been optimized. A small shift of the GTOs when using a finite field method: a) will always lower the energy; b) always increases the energy; c) will increase the energy if the GTOs move in the direction of the field and decrease if they move in the opposite direction; d) will always give a polarizability greater than zero. 6. The magnetic moment M of a particle: a) always has the direction of the particle’s spin angular momentum; b) its length is always an integer or half-integer; c) interacts with a homogeneous magnetic field H, and the interaction energy is equal to 1 2 MH 2 ; d) interacts with a homogeneous magnetic field H, and the interaction energy is equal to −H ·M. 7. If we choose the vector potential A(r) = 1 2 [H ×r],whereH is the magnetic field inten- sity, then: a) we have ∇A =0andA agrees with the Maxwell equations; b) A(r) is a homogeneous field; c) A(r) is directed towards the origin; d) A(r) is parallel to H. 8. The vector potential A(r) of electromagnetic field corresponds to homogeneous mag- netic field H.ThenA: a) is uniquely determined from the Maxwell equation; b) is uniquely determined from the Maxwell equation A =curlH; c) is also a homogeneous field; d) curl[A −∇(x 2 +y 3 +z 4 )]=H. 9. A nuclear shielding constant consists of diamagnetic and paramagnetic parts. Each of these parts: a) represents a second-order effect in perturbation theory; b) represents a first-order effect in perturbation theory; c) changes when the origin of the vector potential A changes; d) represents the Fermi contact term. 10. The London or Gauge-Invariant Atomic Orbital χ L (r −R): a) depends on the vector potential calculated at position R; b) depends on the vector potential calculated at position r; c) does not depend on the vector potential; d) depends on the vector potential at the point shown by r −R. Answers 1d, 2c, 3d, 4a, 5c, 6d, 7a, 8d, 9c, 10a Chapter 13 INTERMOLECULAR INTERACTIONS Where are we? We are already in the crown of the TREE. An example Why does liquid water exist? Why do molecules stick together at low temperatures? Visibly they attract each other for some reason. The interaction is not however very strong since water evaporates when heated (without destroying the water molecules). What is it all about THEORY OF INTERMOLECULAR INTERACTIONS Interaction energy concept () p. 684 • Natural division and its gradation • What is most natural? Binding energy () p. 687 Dissociation energy () p. 687 Dissociation barrier () p. 687 Supermolecular approach () p. 689 • Accuracy should be the same • Basis set superposition error (BSSE) andtheremedy • Good and bad news about the supermolecular method Perturbational approach () p. 692 • Intermolecular distance – what does it mean? • Polarization approximation (two molecules) () • Intermolecular interactions: physical interpretation • Electrostatic energy in the multipole representation and the penetration energy • Induction energy in the multipole representation • Dispersion energy in the multipole representation Symmetry-Adapted Perturbation Theories (SAPT) () p. 710 • Polarization approximation is illegal • Constructing a symmetry adapted function • The perturbation is always large in polarization approximation 681 682 13. Intermolecular Interactions • Iterative scheme of the symmetry adapted perturbation theory • Symmetry forcing • A link to the variational method – the Heitler–London interaction energy • When we do not have at our disposal the ideal ψ A0 and ψ B0 Convergence problems () p. 721 Non-additivity of intermolecular interactions () p. 726 • Many-body expansion of interaction energy • Additivity of the electrostatic interaction • Exchange non-additivity • Induction energy non-additivity • Additivity of the second-order dispersion energy • Non-additivity of the third-order dispersion interaction ENGINEERING OF INTERMOLECULAR INTERACTIONS Noble gas interaction p. 741 Van der Waals surface and radii () p. 742 • Pauli hardness and the van der Waals surface • Quantum chemistry of confined space – the nanovessels Synthons and supramolecular chemistry () p. 744 • Bound or not bound • Distinguished role of the electrostatic interaction and the valence repulsion • Hydrogen bond • Coordination interaction • Hydrophobic effect • Molecular recognition – synthons • “Key-and-lock”, template and “hand-and-glove” synthon interactions Chapter 8 dealt with the question of why atoms form molecules. Electrons and nuclei attract each other, and this results in almost exact neutralization of matter. Despite this, atoms and molecules interact, because • two atoms or molecules cannot occupy the same space, • electrons and nuclei in an atom or molecule may still interact with those in other atoms or molecules. This chapter will tell us about the very reason for this and will give details of the interac- tion. Why is this important? What is the most important message humanity ever learned about matter? According to Richard Feynman the message would be: “The world is built of atoms, which repel each other at short distances and attract at longer ones”. If the intermolecular interactions were suddenly switched off, the world would disintegrate in about a femtosecond, that is in a single period of atomic vibration (the atoms simply would not come back when shifted from their equilib- rium positions). Soon after, everything would evaporate and a sphere of gas, the remainder of the Earth, would be held by gravitational forces. Isn’t it enough? What is needed? • Perturbation theory (Chapter 5, absolutely). Classical works 683 • Variational method (Chapter 5, recommended). • Appendix X, p. 1038 (absolutely). • Many-Body Perturbation Theory (MBPT) (Chapter 10, p. 554, necessary). • Reduced resolvent (Chapter 10, p. 554, necessary). • Appendix Y, p. 1050 (recommended). • Appendix T (mentioned). Classical works Such an important subject was recognized very early. The idea that the cohesion of matter stems from the interaction of small indivisible particles (“atoms”) comes from Democritus. An idea similar to that cited by Feynman was first stated clearly by the Croat scientist Rudjer Boskovi ´ cin“Theoria Philosophiae naturalis”, Venice, 1763. Padé approximants were first proposed in the PhD thesis of Henri Padé entitled “Sur la représen- tation approchée d’une fonction pour des frac- tions rationnelles”, which was published in Annales des Sciences d’Ecole Normale Su- perieure, Suppl. [3], 9 (1892) 1. The role Democritus of Abdera (ca. 460 B.C. – ca. 370 B.C.), Greek philosopher, founder of the first atomic theory. Ac- cording to him, nature repre- sents a constant motion of in- divisible and permanent par- ticles (atoms), whose interac- tions result in various materi- als. It turned out after almost 25 centuries that this hypoth- esis was basically correct! All the written works of Democri- tus have been lost, but his ideas continued to have an important impact on science for centuries. of intermolecular interactions was highlight- ed in the work of Johannes Diderik van der Waals, especially in “Die Kontinuität des gasformigen und flüssigen Zustandes”, Barth, Leipzig (1899, 1900). From that time on, intermolecular interactions are often called van der Waals interactions. The concept of ionic radii was first proposed by Linus Pauling in “The Sizes of Ions and the Struc- ture of Ionic Crystals”, Journal of the Ameri- can Chemical Society, 49 (1927) 765. The Rudjer Josip Boskovi ´ c (1711– 1787), a Croat physicist, math- ematician, astronomer and philosopher from beautiful Dub- rovnik. quantum mechanical explanation of inter- molecular forces, including the ubiquitous dispersion interactions, was given by Fritz London in “Zur Theorie und Systematik der Molekularkräfte”, Zeitschrift für Physik,63 (1930) 245 and in “Über einige Eigenschaften und Anwendungen der Molekularkräfte”from Zeitschrift für Physikalische Chemie (B),11 (1930) 222. Linus Pauling, invited Baker Lecturer to Cornell University, wrote one of the most seminal books in chemistry “The Nature of the Chemical Bond”, Cornell Univ. Press, Ithaca, 1948, where inter alia he eluci- dated the role of hydrogen bonds in forming Johannes Diderik van der Waals (1837–1923), Dutch physicist, professor at the University of Amsterdam. His research topic was the influ- ence of intermolecular forces on the properties of gases (equation of state of the real gas, 1873) and liquids. In 1910 van der Waals received a Nobel Prize “ for his work on the equation of state for gases and liquids ”. structures. The hydrophobic effect was first highlighted by Walter Kauzmann in a paper “Some Factors in the Interpretation of Protein 684 13. Intermolecular Interactions Denaturation”, in Advances in Protein Chemistry, 14 (1959) 1, the effect was further elabo- rated by George Nemethy, Harold Scheraga, Frank Stillinger and David Chandler among others. Resonance interactions were first described by Robert S. Mulliken in an article “The Interaction of Differently Excited Like Atoms at Large Distances”, in Physical Reviews, 120 (1960) 1674. Bogumił Jeziorski and Włodzimierz Kołos extended the existing the- ory of intermolecular forces to intermediate distances (“On the Symmetry Forcing in the Perturbation Theory of Weak Intermolecular Interactions”, International Journal of Quantum Chemistry, 12 Suppl. 1 (1977) 91). THEORY OF INTERMOLECULAR INTERACTIONS There are two principal methods of calculating the intermolecular interac- tions: the supermolecular method and the perturbational method. Both assume the Born–Oppenheimer approximation. 13.1 INTERACTION ENERGY CONCEPT The idea of interaction energy is based on the Born–Oppenheimer (clamped nu- clei, see eq. (6.4)) approximation. Let us define interaction energy at the configu- ration R of the nuclei as E int (R) =E ABC (R) − E A (R) +E B (R) +E C (R) +··· (13.1) where E ABC (R) is the electronic energy (corresponding to E (0) 0 from eq. (6.21)) of the total system, and E A (R), E B (R), E C (R), . are the electronic energies of the interacting subsystems, calculated at the same positions of the nuclei as those in the total system. 13.1.1 NATURAL DIVISION AND ITS GRADATION Although the notion of interaction energy is of great practical value, its theoretical meaning is a little bit fuzzy. Right at the beginning we have a question: interac- tion of what? We view the system as composed of particular subsystems, that once isolated, then have to be put together. For instance, the supersystem may be considered as two interacting water molecules, but even then we still have an uncertainty, whether the two molecules correspond to (I) or to (II): 13.1 Interaction energy concept 685 In addition the system might be considered as composed of a hydrogen molecule interacting with two OH radicals: etc. Choice of subsystems is of no importance from the point of view of mathematics, but is of crucial importance from the point of view of calculations in theoretical chemistry. The particular choice of subsystem should depend on the kind of experiment with which we wish to compare our calculations: • we are interested in the interaction of water molecules when studying water evaporation or freezing; • we are interested in the interaction of atoms and ions that exist in the system when heating water to 1000 ◦ C. Let us stress, that in any case when choosing subsystems we are forced to sin- gle out particular atoms belonging to subsystem 1 A and B. It is not sufficient to define the kind of molecules participating in the interaction, see our examples I and II. If when dividing a system into n subsystems in two ways (I and II), we obtain |E int | I < |E int | II , division I will be called more natural than division II. natural subsystems 13.1.2 WHAT IS MOST NATURAL? Which division is most natural? We do not have any experience in answering such questions. What? Why should we have any difficulties? It is sufficient to consider all possible divisions and to choose the one which requires lowest energy. Unfortu- nately, this is not so obvious. Let us consider two widely separated water molecules (Fig. 13.1.a). 1 This means that the interaction energy idea belongs to classical concepts. In a quantum system, particles of the same kind are indistinguishable. A quantum system does not allow us to separate a part from the system. Despite this, the interaction energy idea is important and useful. . composed of a hydrogen molecule interacting with two OH radicals: etc. Choice of subsystems is of no importance from the point of view of mathematics, but is of crucial importance from the point of. Theory of Weak Intermolecular Interactions”, International Journal of Quantum Chemistry, 12 Suppl. 1 (1977) 91). THEORY OF INTERMOLECULAR INTERACTIONS There are two principal methods of calculating. (1837–1923), Dutch physicist, professor at the University of Amsterdam. His research topic was the influ- ence of intermolecular forces on the properties of gases (equation of state of the real gas, 1873)
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