Chem. Rev. 1988, 88, 899-926 899 Intermolecular Interactions from a Natural Bond Orbital, Donor-Acceptor Viewpoint ALAN E. REED‘ Instltul fur Organische Chemie der Universltat Erlangen-Nurnberg, Henkestrasse 42, 8520 Erlangen, Federal Republic of Germany LARRY A. CURTISS” Chemical Technology Division/Meterials Science and Technology Program, Argonne National Laboratoty, Argonne, Illinois 60439 FRANK WEINHOLD” Theoretical Chemlshy Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706 Received November 10, 1987 (Revised Manuscript Received February 16, 1988) Contents D. Chemisorption 917 E. Relationships between Inter- and 918 I. Introduction 899 Intramolecular Interactions 11. Natural Bond Orbital Analysis 902 IV. Relationship of Donor-Acceptor and 919 A. Occupancy-Weighted Symmetric 902 Electrostatic Models Orthogonalization A. Historical Overview 919 8. Natural Orbitals and the One-Particle 903 B. Relationship to Kitaura-Morokuma Analysis 920 Density Matrix C. Semiempirical Potential Functions 92 1 C. Atomic Eigenvectors: Natural Atomic 904 V. Concluding Remarks 922 Orbitals and Natural Population Analysis D. Bond Eigenvectors: Natural Hybrids and 904 Natural Bond Orbitals E. Natural Localized Molecular Orbitals 905 1. Introdud/on F. Hyperconjugative Interactions in NBO 906 The past 15 years has witnessed a golden age of Analysis discovery in the realm of “van der Waals chemistry”. II I. Intermolecular Donor-Acceptor Models Based 906 The vm der wads bonding regimelies at the interface on NBO Analysis between two well-studied interaction types: the A. H-Bonded Neutral Complexes 906 short-range, strong (chemical) interactions of covalent 1. Water Dimer 906 type, and-the long-range, weak (physical) interactions 2. OC.-HF and COv-HF 908 of dispersion and multipole type. It was therefore surprising to discover that this borderline region gives Complexes of NO and HF 909 3. rise to species whose structural and energetic patterns 4. Of a Large Series Of H-Bond 910 are distinctivelv novel. As exDerimental nozzle beam14 Complexes and ab initio ~omp~tational*studies~*~have combined 5. Cooperativity in H-Bonding 911 to reveal the rich architectural patterns of van der 6. Hydrophobic Interactions 912 Waals bonding, efforts have been made to extend the 7. Other Tooics 913 elementary principles of valence theory to encompass B. Non-H-Bonded Neutral Complexes 914 such bonds within a general conceptual framework of covalent and noncovalent interactions. The present 1. Survey of a Large Series of 914 article reviews progress that has been made toward this Non-H-Bonded Complexes goal by the method of natural bond orbital analysis, 2. Competition between H-Bonded and 914 particularly for H-bonded and other strongly bound van Non-H-Bonded Structures der Waals complexes. 3. Complexes of Rare Gases with Be0 915 Natural bond orbital (NBO) analysis originated’ as C. Ion-Molecule Complexes and Contact Ion 915 a technique for studying hybridization and covalency Pairs effects in polyatomic wave functions, based on local 1. Anion-Water Complexes 915 block eigenvectors of the one-particle density matrix (see section 11). NBOs were conceived as a “chemist’s 2. Cation-Water Complexes 916 basis set” that would correspond closely to the picture 3. Bifluoride Ion 916 of localized bonds and lone pairs as basic units of mo- 4. “Salt” Isomer of Carbon Tetrachloride 916 lecular structure. The NBO for a localized u bond uAB 5. Ground and Excited States of (CO),’ 916 between atoms A and B is formed from directed or- 0009-2665/88/0788-0899$06.50/0 @ 1988 American Chemical Society 900 Chemical Reviews. 1988, VOI. 88, NO. 6 Reed et 81. Alan E. Reed was born in Chicago, IL, in 1958 and received his B.S. degree In chemistry from the University of Illinois at Champ- Frank A. Welnhokl was bwn in Scottsbluff. NE. in 1941. He SMied aign-Urbana In 1980, having carried out experimental research chemistry as an undergraduate at the University of Colorado. both there and at the University of Chicago. He received his h.D. Boulder (1958-1962), as a Fulbright Scholar at the Universitat in physical chemistry from the University of Wisconsin at Madison FreiburglBreisgau in Germany (1962-1963), and as a graduate in 1985. working with Prof. Frank Weinhold, a part of this work student (under Prof. E. B. Wilson, Jr.) at HaNard University. where being carried out with Dr. Larry Curtiss at Argonne National Lab- he received the Ph.D. degree in physical chemistry in 1967. His oratory. Curremly. he is doing postdoctoral research with Prof. Paul postdoctoral studies were at Oxford (with Prof. C. A. Coulson) and von Ragu6 Schleyer at Erlangen. West Germany. His scientific Berkeley (Miller Fellow). He has taught at Stanford University and interests concern the rich range of diversity and possibility in the University of Wisconsin, Madison, where he is currently Pro- chemical Structure and bonding, from solid state to biochemical fessor of Chemistry and Director of the Theoretical Chemistry systems, and have focused primarily on the theoretical description Instiiute. His principal research interests include upper and lower of inter- and intramolecular bonding involving elements H to Ar, bounds for quantum mechanical properties, metric geometry of including studies of hypervalency. hyperconjugation. and the equilibrium thermodynamics. and natural bond orbital studies of anomeric effect. covalent and noncovalent interactions. atomic and bond properties. Ab initio wave functions transformed to NBO form are found to be in good agreement with Lewis structure concepts and with the basic Pauling-Slater-Coulson picture” of bond hy- bridization and polarization. The filled NBOs um of the “natural Lewis structure” me therefore well adapted to describing covalency effects in molecules. However, the general transformation to NBOs also leads to orbitals that are unoccupied in the formal Lewis structure and that may thus be used to describe non- covalency effects. The most important of these are the antibonds u*ml* Larry A. Curtiss was born in Madison. WI. in 1947 and received his B.S. degree in chemistry from the University of Wisconsin in U*AB= CBhA - CAhB Ob) 1969. He received his Ph.D. degree in physical chemistry in 1973 wofking with Professw John A. Popb. He tbn spent about 2 years which arise from the same set of atomic valence-shell as a httelle lnstiute Fellow at Banelle Memurial Institute (wofking hybrids that unite to form the bond functions um, eq with Professor C. W. Kern) in Columbus. OH. In 1976 he joined la. The antibonds represent unused valence-shell ca- the Chemical Technology Division at Argonne National Laboratory. His research interesls include theoretical studies of van der Waals pacity, spanning portions of the atomic valence space complexes. ordered and ionic solutions, zeolies, hetereogeneous that are formally unsaturated by covalent bond for- charge-transfer reactions, and small molecules and their cations. mation. Small occupancies of these antibonds corre- In addition, he has been involved in experimental studies of gas- spond, in Hartree-Fock theory, to irreducible depar- phase association reactions. tures from the idealized Lewis picture and thus to small noncovalent corrections to the picture of localized co- valent bonds. thonormal hybrids hA, hB [natural hybrid orbitals The energy associated with the antibonds can be (NHOs)l numerically assessed by deleting these orbitals from the basis set and recalculatine the t~talenem to determine um = cAhA+ cBhB (la) the associated variationai energy 1oweri;g. In this way and the natural hybrids in turn are composed from a one obtains a decomposition of the total energy E into set of effective valence-shell atomic orbitals [natural components associated with covalent (Eoo= Em)and atomic orbitals (NAOS)],~*~optimized for the chosen noncovalent (Eo8 = Enon.Le+)contributions wave function. A distinguishingfeature of such natural E = E,,,, E., (2) localized functions (analogous to classic “natural + orbitals” in the Lowdin delocalized senselo) is the si- NBO decompositions of this form have now been ob- multaneous requirement of orthonormality and maxi- tained for a large number of closed-shell and open- mum occupancy, leading to compact expressions for shelP molecular species. In equations such as (1) and Donor-Acceptor Interactlons Chemical Reviews, 1988, Vol. 88, No. 6 901 TABLE I. NBO Energy Decompositions for Selected E2 /- \ Molecules (RHF/6-31G* Level, Pople-Gordon Idealized \ Geometry), Showing the Covalent Contribution E(Lewis) = I I \ E,,, the Noncovalent Contribution E(non-Lewis) = E,,., -%* and the Percentage Contribution % E(Lewis) Associated I I with the Natural Lewis Structure” I I I I molecule E(Lewis) E(non-Lewis) % E(Lewis) I I BH, -26.384470 -0.005434 99.979 I I CH; -40.187329 -0.007732 99.981 NH3 -56.180217 -0.003527 99.994 HZO -76.007041 -0.002827 99.996 HF -100.001908 -0.000900 99.999 A1Ha -243.570589 -0.045162 99.982 SiH, -291.192327 -0.032763 99.989 Figure 1. Perturbative donor-acceptor interaction, involving a PH3 -342.410798 -0.019140 99.994 filled orbital u and an unfilled orbital u*. H2S -398.652368 -0.007483 99.998 HCl -460.056952 -0.002661 99.999 CFaH -336.469129 -0.295778 99.912 MOs are strictly unoccupied and thus play no role in C2H6 -79.170183 -0.057562 99.927 the wave function or any observable property, whereas C2HI -77.953649 -0.076717 99.902 the antibonds (u*-) generally exhibit nonzero occu- CzHz -76.728306 -0.089021 99.884 HZCO -113.762634 -0.101071 99.911 pancies, and their contibutions lead to definite energy c&6 -230.107441 -0.594436 99.742 lowerings and changes in the form of the wave function. The role of antibonds can be seen by transforming the ONote the effect of vicinal u - u* interactions (molecules 11-15 of the list) and of aromatic “resonance” (C6He)in reducing the occupied canonical MOs to localized molecular orbital dominance of a sinele Lewis structure.