JCP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

You Tube Flickr Twitter UniPHY Group iResearch App Facebook

Journal of Chemical Physics / Volume 135 / Issue 24 / ARTICLES / Surfaces, Interfaces, and Materials
FREE

FULL-TEXT OPTIONS:

J. Chem. Phys. 135, 244701 (2011); http://dx.doi.org/10.1063/1.3660673 (22 pages)

Thermal fluctuations in shape, thickness, and molecular orientation in lipid bilayers

Max C. Watson1, Evgeni S. Penev2, Paul M. Welch3, and Frank L. H. Brown1,2

1Department of Physics, University of California, Santa Barbara, California 93106, USA
2Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, USA
3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

View MapView Map

(Received 19 July 2011; accepted 26 October 2011; published online 22 December 2011)

We present a unified continuum-level model for bilayer energetics that includes the effects of bending, compression, lipid orientation (tilting relative to the monolayer surface normal), and microscopic noise (protrusions). Expressions for thermal fluctuation amplitudes of several physical quantities are derived. These predictions are shown to be in good agreement with molecular simulations.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. BILAYER ENERGETICS
    1. GEOMETRIC DESCRIPTION
    2. Macroscopic bilayer free energy
    3. Protrusions
  3. FLUCTUATION SPECTRA OF HOMOGENEOUS MEMBRANES
    1. Decoupled protrusion/bending approximation
    2. γλ = 0 approximation
  4. SIMULATION DETAILS
    1. Implicit solvent model
    2. MARTINI force field model (DPPC)
  5. COMPARISON TO THEORY
  6. CONCLUSION

RELATED DATABASES

Scopus

View This Record in Scopus

Web of Science

View this record in Web of Science

View Web of Science related articles

KEYWORDS and PACS

Keywords

bending, biomembranes, biothermics, cellular biophysics, compressibility, lipid bilayers, molecular biophysics, monolayers

PACS

  • 87.19.Pp

    Biothermics and thermal processes in biology

  • 87.16.dj

    Dynamics and fluctuations

  • 87.16.dm

    Mechanical properties and rheology

  • 87.16.dt

    Structure, static correlations, domains, and rafts

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3660673

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

  1. E. Browicz, Zbl. Med. Wiss. 28, 625 (1890).
  2. F. Brochard and J. F. Lennon, J. Phys. (France) 36, 1035 (1975). [Inspec] [ISI]
  3. J. Israelachvili and H. Wennerström, J. Phys. Chem. 96, 520 (1992).
  4. R. Lipowsky, Nature (London) 349, 475 (1991). [MEDLINE]
  5. C. S. Peskin, G. M. Odell, and G. F. Oster, Biophys. J. 65, 316 (1993). [MEDLINE] [CAS]
  6. R. Bruinsma and P. Pincus, Curr. Opin. Solid State Mater. Sci. 1, 401 (1996). [Inspec] [ISI]
  7. R. Golestanian, M. Goulian, and M. Kardar, Phys. Rev. E 54, 6725 (1996). [MEDLINE] [CAS]
  8. W. Helfrich, Z. Naturforsch. C 28, 693 (1973). [MEDLINE]
  9. P. B. Canham, J. Theor. Biol. 26, 61 (1970). [MEDLINE]
  10. F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
  11. R. Lipowsky and S. Grotehans, Biophys. Chem. 49, 27 (1994). [ISI]
  12. J. N. Israelachvili, Intermolecular and Surface Forces (Academic, New York, 1985).
  13. F. C. MacKintosh and T. C. Lubensky, Phys. Rev. Lett. 67, 1169 (1991). [MEDLINE]
  14. M. Hamm and M. M. Kozlov, Eur. J. Phys. E. 3, 323 (2000). [ISI]
  15. Y. Kozlovsky and M. M. Kozlov, Biophys. J. 82, 882 (2002). [ISI] [MEDLINE] [CAS]
  16. P. I. Kuzmin, S. A. Akimov, Y. A. Chizmadzhev, J. Zimmerberg, and F. S. Cohen, Biophys. J. 88, 1120 (2005). [Inspec] [MEDLINE]
  17. A. Kessel, N. Ben-Tal, and S. May, Biophys. J. 81, 643 (2001). [Inspec] [ISI] [MEDLINE] [CAS]
  18. G. Khelashvili, G. Pabst, and D. Harries, J. Phys. Chem. B 114, 7524 (2010).
  19. J. Aittoniemi, T. Róg, P. Niemelä, M. Pasenkiewicz-Gierula, M. Karttunen, and I. Vattulainen, J. Phys. Chem. B 110, 25562 (2006).
  20. S. May and A. Ben-Shaul, Phys. Chem. Chem. Phys. 2, 4494 (2000).
  21. S. May and A. Ben-Shaul, Biophys. J. 76, 751 (1999). [Inspec] [ISI] [MEDLINE] [CAS]
  22. S. May, Eur. Biophys. J. 29, 17 (2000). [Inspec] [MEDLINE]
  23. S. May, Langmuir 18, 6356 (2002). [ISI] [CAS]
  24. M. Fosňarič, A. Iglič, and S. May, Phys. Rev. E 74, 051503 (2006).
  25. K. Bohinc, V. Kralj-Iglič, and S. May, J. Chem. Phys. 119, 7433 (2000)JCPSA6000119000014007435000001.
  26. Y. Kozlovsky, J. Zimmerberg, and M. M. Kozlov, Biophys. J. 87, 999 (2004). [MEDLINE]
  27. J.-B. Fournier, Eur. J. Phys. E 11, 261 (1999).
  28. J.-B. Fournier, Europhys. Lett. 43, 725 (1998). [Inspec] [ISI]
  29. B. West, F. L. H. Brown, and F. Schmid, Biophys. J. 96, 101 (2009). [Inspec] [MEDLINE] [CAS]
  30. M. Hamm and M. M. Kozlov, Eur. J. Phys. E 6, 519 (1998). [ISI]
  31. S. Tristram-Nagle, R. Zhang, R. M. Suter, C. R. Worthington, W. J. Sun, and J. F. Nagle, Biophys. J. 64, 1097 (1993).
  32. T. C. Lubensky and F. C. MacKintosh, Phys. Rev. Lett. 71, 1565 (1993). [MEDLINE]
  33. C.-M. Chen, T. C. Lubensky, and F. C. MacKintosh, Phys. Rev. E 51, 504 (1995). [MEDLINE]
  34. K. Sengupta, V. A. Raghunathan, and Y. Hatwalne, Phys. Rev. Lett. 87, 055705 (2001). [ISI] [MEDLINE] [CAS]
  35. U. Seifert, J. Shillcock, and P. Nelson, Phys. Rev. Lett. 77, 5237 (1996). [MEDLINE]
  36. E. A. G. Aniansson, S. N. Wall, M. Almgren, H. Hoffmann, I. Kielmann, W. Ulbricht, R. Zana, J. Lang, and C. Tondre, J. Phys. Chem. 80, 905 (1976). [ISI] [CAS]
  37. T. J. McIntosh, S. Advani, R. E. Burton, D. V. Zhelev, D. Needham, and S. A. Simon, Biochemistry 34, 8520 (1995).
  38. A. Halperin and O. Mouritsen, Eur. Biophys. J. 34, 967 (2005).
  39. R. Goetz, G. Gompper, and R. Lipowsky, Phys. Rev. Lett. 82, 221 (1999).
  40. E. Lindahl and O. Edholm, J. Chem. Phys. 113, 3882 (2000)JCPSA6000113000009003882000001.
  41. E. Lindahl and O. Edholm, Biophys. J. 79, 426 (2000). [Inspec] [ISI] [MEDLINE] [CAS]
  42. S. J. Marrink and A. E. Mark, J. Phys. Chem. 105, 6122 (2001). [CAS]
  43. S. W. Chiu, S. Vasudevan, E. Jakobsson, R. J. Mashl, and H. L. Scott, Biophys. J 85, 3624 (2003).
  44. G. Brannigan and F. Brown, Biophys. J. 90, 1501 (2006). [Inspec] [MEDLINE] [CAS]
  45. G. Brannigan, P. F. Philips, and F. L. H. Brown, Phys. Rev. E 72, 011915 (2005). [ISI]
  46. G. Brannigan, A. C. Tamboli, and F. L. H. Brown, J. Chem. Phys. 121 (2004)JCPSA6000121000007003259000001. [ISI] [MEDLINE]
  47. M. J. Stevens, J. Chem. Phys. 121, 11942 (2004)JCPSA6000121000023011942000001. [ISI] [MEDLINE]
  48. I. R. Cooke, K. Kremer, and M. Deserno, Phys. Rev. E 72, 011506 (2005).
  49. W. K. den Otter, J. Chem. Phys. 123, 214906 (2005)JCPSA6000123000021214906000001. [MEDLINE]
  50. J. Neder, B. West, P. Nielaba, and F. Schmid, J. Chem. Phys. 132, 115101 (2009)JCPSA6000132000011115101000001. [MEDLINE]
  51. B. West and F. Schmid, Soft Matter 6, 1275 (2010).
  52. E. R. May, A. Narang, and D. I. Kopelevich, Phys. Rev. E 76, 021913 (2007). [ISI]
  53. E. R. May, A. Narang, and D. I. Kopelevich, Mol. Simul. 33, 787 (2007).
  54. F. M. Thakkar, P. K. Maiti, V. Kumaran, and K. G. Ayappa, Soft Matter 7, 3963 (2011).
  55. S. J. Marrink, J. H. Risselada, S. Yefimov, P. D. Tieleman, and A. H. de Vries, J. Phys. Chem. B 111, 7812 (2007). [Inspec] [MEDLINE]
  56. S. J. Marrink, A. H. de Vries, and A. E. Mark, J. Phys. Chem. B 108, 750 (2004).
  57. I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (Taylor & Francis, London, 2004).
  58. A. Imparato, J. Chem. Phys. 124, 154714 (2006)JCPSA6000124000015154714000001. [MEDLINE]
  59. J.-B. Fournier and C. Barbetta, Phys. Rev. Lett. 100, 078103 (2008).
  60. W. Cai, T. C. Lubensky, P. Nelson, and T. Powers, J. Phys. II 4, 931 (1994). [ISI] [CAS]
  61. O. Farago and P. Pincus, Eur. Phys. J. E 11, 399 (2003). [MEDLINE]
  62. O. Farago and P. Pincus, J. Chem. Phys. 120, 2934 (2004)JCPSA6000120000006002934000001. [MEDLINE]
  63. J. Stecki, J. Chem. Phys. 125, 154902 (2006)JCPSA6000125000015154902000001. [MEDLINE]
  64. F. Schmid, Eur. Phys. Lett. 95, 28008 (2011).
  65. S. A. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes (Westview Press, Boulder, CO, 1994).
  66. N. G. V. Kampen, Stochastic Processes in Physics and Chemistry (Amsterdam, North Holland, 2007).
  67. S. J. Plimpton, J. Comput. Phys. 117, 1 (1995).
  68. R. Goetz and R. Lipowsky, J. Chem. Phys. 108, 7397 (1998).
  69. P. J. Steinbach and B. B. Brooks, J. Comput. Chem. 15, 667 (1994).
  70. O. Farago, J. Chem. Phys. 119, 596 (2003).
  71. D. Marsh, Biochim. Biophys. Acta. 1286, 183 (1996).
  72. E. Lindahl and O. Edholm, J. Chem. Phys. 113, 3882 (2000)JCPSA6000113000009003882000001.
  73. M. Bhandarkar et al., NAMD User's Guide. Version 2.7b2, University of Illinois and Beckman Institute, Urbana, IL (2009), sec. 12.2.
  74. A. Harasima, Adv. Chem. Phys. 1, 203 (1958).
  75. D. van der Spoel, E. Lindahl, B. Hess, G. Groenhof, A. E. Mark, and H. J. C. Berendsen, J. Comput. Chem. 26, 1701 (2005). [Inspec] [MEDLINE] [CAS]
  76. H. J. C. Berendsen, D. van der Spoel, and R. van Drunen, Comput. Phys. Commun. 91, 43 (1995).
  77. B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, J. Chem. Theory Comput. 4, 435 (2008).
  78. See http://md.chem.rug.nl/cgmartini to download pre-assembled bilayer coordinates and force field files.
  79. S. O. H. Ollila, J. H. Risselada, M. Louhivuori, E. Lindahl, I. Vattulainen, and S. J. Marrink, Phys. Rev. Lett. 102, 078101 (2009). [MEDLINE]
  80. J. H. Irving and J. G. Kirkwood, J. Chem. Phys. 18, 817 (1950)JCPSA6000018000006000817000001.
  81. J. Sonne, F. Y. Hansen, and G. H. Peters, J. Chem. Phys. 122, 124903 (2005)JCPSA6000122000012124903000001. [ISI] [MEDLINE]
  82. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, England, 2007).
  83. S. E. Feller and R. W. Pastor, J. Chem. Phys. 111, 1281 (1999)JCPSA6000111000003001281000001.
  84. S. Shkulipa, Computer Simulations of Lipid Bilayer Dynamics, Ph.D. dissertation, University of Twente, The Netherlands, 2006).
  85. Q. Waheed and O. Edholm, Biophys. J. 97, 2754 (2009). [Inspec] [MEDLINE] [CAS]
  86. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Dover, New York, 2003).
  87. G. Brannigan and F. L. H. Brown, Biophys. J. 92, 864 (2007). [MEDLINE]
  88. M. Watson and F. L. H. Brown, Biophys. J 98, L9 (2010). [MEDLINE] [CAS]
  89. E. G. Brandt, A. R. Braun, J. N. Sachs, J. F. Nagle, and O. Edholm, Biophys. J. 100, 2104 (2011).
  90. S. May, Y. Kozlovsky, A. Ben-Shaul, and M. M. Kozlov, Eur. Phys. J. E 14, 299 (2004). [MEDLINE]
  91. M. Frigo and S. G. Johnson, Proccedings of IEEE 93, 216 (2005).

Figures (15) Tables (3)

Figures (click on thumbnails to view enlargements)

FIG.1
For each lipid, dark gray circles mark the portion of the molecule separating the polar head from the hydrocarbon tails. On a coarse-grained level, the polar-nonpolar interfaces are described by z(1) and z(2). The unit vectors math(α) are normal to z(α) and point toward the interior of the bilayer. The unit vectors field math(α) points along the hydrocarbon chains. b(α)math(α) extend from z(α) to the surface separating the two leaflets, z(m). In other words, the top monolayer is bounded by z(1) and z(m), while the bottom monolayer is bounded by z(m) and z(2). The mean height z+ is the average of z(1) and z(2). Left: a bilayer in its minimal energy configuration, in which λ(α) = 0, z(m) = z+, N(α) = n(α) = 0, the thickness is 2b0 and the area per molecule is Σ0 (dotted red). The volume per lipid v satisfies v = b0Σ0. Since there are no protrusions, h(α) = z(α). Right: an arbitrarily deformed bilayer. On short length scales, the polar-nonpolar interfaces h(α) are not smooth (dashed curves). The protrusion fields λ(α) displace the interface in the normal direction, so that (−1)αλ(α)math(α) (dashed vectors) extend from z(α) to h(α). The fields z(m) (black) and z+ (blue) differ in general. The thickness deformations are exaggerated for illustrative purposes. We will assume throughout the paper that the absolute values of the following quantities are much less than one: Nj(α), nj(α), ∇N(α), ∇n(α), ∇z(m), Σ(α)/Σ0−1, b(α)/b0−1, |z(α)−z(m)|/b0−1, (z+−z(m))/b0,∇λ(α),λ(α)/b0.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Top: The hydrocarbon-water interface of the middle lipid is protruded upwards (λ(1) > 0). The measured director math(1) points from the displaced surface toward the end of the hydrocarbon chain (along the red dashed vector). The dashed vector only shows the direction of math(1) but not its magnitude, since math(1) is a unit vector. The monolayer thickness is large compared to λ(1), so math(1) ≈ math(1). Bottom: math(1) ≠ math(1) due to a gradient in the protrusion field.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
The starting point of our model is the phenomenological macroscopic free energy per lipid math(α),21 , 22 which is a function of the hydrocarbon chain length b(α), the cross sectional area at the polar-nonpolar interface Σ(α) (green), and the cross sectional area at the center of the head group Σh(α) (yellow). The two surfaces which contain Σ(α) and Σh(α) are separated by a fixed distance ℓh. The volume v of the hydrocarbon chain region (finely dashed box) is constant to enforce chain incompressibility.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Here, two neighboring rows of lipids have different values of m(1). Twist is present since the tilt vectors point in the x direction, but vary in the y direction. Thus, the κθ and κtw terms in Eq. ( 13 ) are nonzero.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.5
Snapshot from our implicit solvent model (CG). Each lipid consists of five beads (see Fig. 7).

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.6
The stress profile within the bilayer for our implicit solvent model (CG). The troughs at ±3 nm represent the repulsion between the head beads, while the peaks at ±2 nm are due to the strong attraction between interface beads.71 The monolayer spontaneous curvature c0 is related to s(z) through Eq. ( 47 ). The stress profile we measured for the MARTINI model of DPPC is the same as in the original paper.55

FIG.6 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.7
A schematic diagram of the force fields used in the two lipid systems we studied. We also illustrate how the director math(α) is measured within each model. Note that for visual purposes math(α) is not normalized. Left: Our implicit solvent model (CG). Each lipid consists of a hydrophilic head bead (green), an interfacial bead (orange), and three hydrophobic tail beads (gray). Though the potentials were designed to mimic generic intermolecular forces, no explicit electrostatic interactions are present. math(α) points from the interfacial bead toward the last tail bead. Right: For the MARTINI force field (DPPC), each lipid consists of a positively charged bead representing the choline group (red), a negatively charged bead representing the phosphate group (purple), two beads of intermediate hydrophobicity representing the glycerol ester linkage (green), and two chains of four hydrophobic beads each, representing the hydrocarbons (brown). The charged beads interact via a shifted Coulombic potential energy function.55 math(α) points from the midpoint of the interfacial beads toward the midpoint of the last tail beads.

FIG.7 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.8
Snapshot from the MARTINI force field model of DPPC. Each lipid consists of 12 beads (see Fig. 7). Solvent particles are not shown.

FIG.8 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.9
Height 〈|hq|2〉, thickness 〈|tq|2〉, and tilt 〈|mathq⊥|2〉, 〈|mathq⊥|2〉, 〈|mathq∥|2〉, 〈|mathq∥|2〉 fluctuations for the CG model. Simulation data are displayed as circles. The data points used for the fits (black) are evenly spaced along the qx and qy axes. Data points which do not lie on the axes are shown in orange. The solid curves represent best fits of the data to Eqs. ( 39 , 40 , 41 , 42 , 43 ) with γλ = 0. Fit parameters are listed in Table 3. 〈|hq|2〉 is also plotted on a semi-log scale in the inset. Dashed curves correspond to the protrusion-free limit by using Eqs. ( 39 , 40 , 41 , 42 ) and the same values in Table 3 but with kλ → ∞. While only a small contribution to the height and thickness modes, protrusions are essential for explaining the qualitative behavior of 〈|mathq∥|2〉 and 〈|mathq∥|2〉.

FIG.9 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.10
Fluctuations and fits to the DPPC model. See Fig. 9 for details.

FIG.10 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.11
Though not used in the fitting procedure, the data for 〈hqmath−∥q〉, 〈tqmath−∥q〉 and 〈|mathq(α)|2〉 agree with the theoretical predictions very closely. Both the theory and data show that the fluctuations in the cross-terms are purely imaginary. This may be understood mathematically since mathq∥* = −math−q∥ and hq* = h−q and similarly for the peristaltic term (see Sec. 3). The monolayer tilt averaged over all directions 〈|mathq(α)|2〉 is also plotted in MNK.

FIG.11 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.12
The upper bounding surface of dV(1) is z(1) (green), which contains math1, math2, math3, and math4.

FIG.12 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.13
The infinitesimal volume element dV(1). The bottom vertices (math5,math6,math7,math8) are located at the end of the lipid tails whose polar-nonpolar interfaces lie at (math1,math2,math3,math4), respectively. For graphical clarity we write x′ ≡ x + dx and y′ ≡ y + dy. z(1) is shown in green.

FIG.13 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.14
−ℓhmath(1) extends from z(1) (green) to the head group surface (yellow). The coloring corresponds to Fig. 3.

FIG.14 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.15
Schematic of lipid assignment to a M × M grid. Filled circles represent lipid interface points (brighter: lipid in the lower monolayer; darker: lipid in upper monolayer). The highlighted square is an example of a patch that does not contain lipids in the upper monolayer (njk(1) = 0).

FIG.15 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

Tables

Table I. The list of fields associated with the bilayer along with the equation or figure where they are first defined. The superscript α = {1, 2} refers to the top and bottom leaflets, respectively. Though the list is long, the only fields present in the final expression for the bilayer free energy (Eqs. (26,27,28)) are {z±, λ±, math, math}.

View Table
Table II. List of symbols along with the equation or figure where they are defined. Note that all elastic constants are defined as monolayer properties and that we define c0 to be the spontaneous total curvature of the monolayer and choose its sign such that c0 > 0 implies lipids with a preference for micelle over reverse micelle geometries. These conventions agree with some works and differ from others; extreme care must be exercised in comparing to prior results. (In particular, the conventions used in BB differ from those introduced here for the quantities kA, kcb, and c0. In BB, kA and kc refer to bilayer properties and correspond to double the values defined here. c0 in BB is defined as the monolayer mean curvature and corresponds to half the value defined here.)

View Table
Table III. The material parameters for our coarse-grained implicit solvent model (CG) and the MARTINI force field simulation (DPPC). The values of {kλ, kc, kA/b02, Ω/b0, κθ,κ tw } are the best fit values to the thermal fluctuation spectra. The 95% confidence intervals are written in parenthesis. The monolayer thickness b0 for CG and DPPC is 2.4 nm and 1.8 nm, respectively. A value of kA was also extracted by measuring the variance in the simulation box size, Eq. (46). All constants are defined with respect to the monolayer and c0 is defined as the monolayer total spontaneous curvature. Care must be taken when comparing these quantities to earlier studies where conventions sometimes differ (see Table 2 for a discussion of the conventions used in BB). c0 was calculated by using the best fit value for kc combined with the first moment of the stress profile, Eq. (47).

View Table

Close

Google Calendar

ADVERTISEMENT

Your Recent History Recent History Help

Recently Viewed

  1. Thermal fluctuations in shape, thickness, and molecular orientation in lipid bilayers
  2. Accurate ab initio quartic force fields of cyclic and bent HC2N isomers
  3. Theoretical characterization of intermolecular vibrational states through the multi-configuration time dependent Hartree approach: The He2,3ICl clusters
  4. The ab initio ground-state potential energy function of beryllium monohydride, BeH
  5. Long-range interactions between like homonuclear alkali metal diatoms
Go to AIP Home
Copyright © American Institute of Physics
close