ionic liquids: from static quantum chemical calculations ... · frank jensen wiley & sons 2nd...

Ionic liquids: From static quantumchemical calculations to simulations

Barbara Kirchnerwww.uni-leipzig.de/~quant

2

Outline

Introduction

Calculations: Static calculations

AIMD simulations

MD simulations

Summary

3

Introduction

Introduction


AIMD simulations

MD simulations

Summary

4

Ionic liquids (IL) are salts that are liquid at room temperature

ILs consist of organic cations (low symmetry) and weakly basic (in)organicanions (diffuse charge)

Seddon “To use the term molten salts to describe these novel systems (BK: ILs) is as archaic as describing a car as a horseless carriage” Seddon, J. Chem. Tech. Biotechnol. 1997, 68, 351-356

Origin, see also textbook “An Introduction to Ionic Liquids“ by Michael Freemantle

S. Gabriel 1888: Ethanolammonium nitrate mp: 52-55 °C

P. Walden 1914: Ethylammonium nitrate mp: 12.5 °C

What are ionic liquids?

5

Why are IL intersting and problematic?

MacFarlane: “While early work in the field

tended to presume that ILs hadvery similar properties as a

class, it is now widelyrecognized that, in fact, they

offer a wide range of properties and that one of the

only properties that can bethought of as ubiquitous amongILs is ion conductivity.“ MacFarlane,

Top. Curr. Chem. “Ionic liquids” ed. B. Kirchner, 2010

MolecularMolecular Dynamics (MD)Dynamics (MD)HugeHuge systemssystems: : ClassicalClassical MechanicsMechanics

Ab Ab initioinitio MD (AIMD)MD (AIMD)Large sytems

III RMFr&&

r=

{ }[ ]Iii

ii

N

III RRM

r&&

r& ;

21

1

2 φεφφμ −+∑∑=

Quantum Quantum ChemistryChemistry (QC)(QC)Small Small systemssystems: Quantum : Quantum mechanicsmechanics

Quantum Cluster Quantum Cluster EquilibriumEquilibrium (QCE)(QCE)

Different Different clustersclusters and and statisticalstatistical mechanicsmechanics

εSCFC=

)1/()/(

−⎥⎦

⎤⎢⎣

⎡= kj

kjii

A

ii

k

kjj N

qnqn

Methods we apply to ILs

7

Scale-transfer modeling

8

Quantum Chemistry *Important steps:- After Born-Oppenheimer approximationSchrödinger equation for electronic problem:

What is described:• static picture• single molecules• Quantum mechanics• CPU/Duration: 1 week-1montht• Usual PC (large memory)

Useful Literature:- Introduction to Computational ChemistryFrank Jensen Wiley & Sons 2nd ed. 2007- Quantum Chemistry; Ira N. Levine, Prentice Hall; 5th Ed.- Modern Quantum; Attila Szabo, Neil S. Ostlund; Dover Pubn Inc (1996)

[ ] Ψ=Ψ++=Ψ EVVTH eeeKeelˆˆˆˆ

Approach for Ψ=Ψ(1,2,3,4,…):- Slater-Determinant (Pauli principle)- Next apply Variation principle- Obtain the Hartree-Fock equation:

[ ] ij

ijiKJh χεχ ∑=++ ˆˆˆ

Ĵ (Coulomb-OP) and K (Exchange-OP) contain the unknown spin orbitals χ→ Iterative solution of self-consistent field equations

9

Density functional theory *

Crucial for performance: exchange-correlation (XC) functional; Generations:1. L(S)DA: Local spin density; depends only on ρ (ok for solid-state physics but

fail for chemistry)2. GGA: Generalized gradient approximation; depends also on ∇ρ (ok for

thermochemistry but barriers heights)3. Meta-GGA: kinetic energy density is added τ (performance slightly better

than GGA)4. Hybrid functionals (B3LYP): add Hartree-Fock exchange (exact exchange);

(energetics better, but barrier heights are still underestimated; Non-covalent interactions are not well described, transition metals not good enough)

(Zhao and Truhlar, Acc. Chem. Res. 41 2008, 157-167)

Important idea:Obtain energy from density which is a function of only three variables: ρ(r) or n(r)

][][][][][ ρρρρρ xcsext VTJVE +++=

10

Molecular dynamics simulations *Basics within computer:1. Store start positions and start velocities of all atoms2. Calculate from intermolecular potential (Newton) new positions and

velocities3. Move the atoms to new positions

Force to move the atoms: From derivative of the potential with respect to the coordinates

I

II R

RUF r

rr

∂∂

−=)(

Newton: Relation between acceleration and force

II FRMrr

&& =

Numerical integration: step by step (Taylor expansion):

Δt: discrete time step, e.g. 1 fs

)...(61.0)(5.0)()()( 32 tBttRttRttRttR IIIInewI Δ+Δ+Δ+=Δ+

r&&

r&

rr

)...(5.0)()()( 2 tBttRttRttR IIInewI Δ+Δ+=Δ+

r&&

r&

r&

Useful Literature:Computer

Simulation of Liquids; M. P. Allen,

D. J. Tildesley; Oxford University

Press; (1989)

UnderstandingMolecular

Simulation: FromAlgorithms to

Applications; D. Frenkel, B. J. Smit

Academic Press

11

Molecular dynamics simulations II *

What is described:- Dynamic in atom coordinates; electrons indirectly via the potential- Many particles (>100000)- classical mechanics- CPU: 1 week-1month- Large storage for trajectory

Velocity Verlet algorithm:

]2/[)()()()( 2IIII

newI MtFttRttRttR

rr&

rrΔ+Δ+=Δ+

]2/[)]()([)()( IIIInewI MtFttFttRttR

rrr&

r& +Δ+Δ+=Δ+

Useful Literature:- Computer Simulation of Liquids; M. P. Allen, D. J. Tildesley; Oxford University Press; (1989)- Understanding Molecular Simulation: From Algorithms to Applications; D. Frenkel, B. J. Smit, Academic Press

12

Ab initio Molecular dynamics simulations *Important idea: Calculate the forces or the potential on the flyEhrenfest (1927), Dirac (1930): Theory of time-dependent self consistent field equations for nuclear and electronic motion

Born-Oppenheimer MD simulationsInteraction energy U(RI) is electronic ground state energy:

Lagrangian:

Within computer:1. Electronic structure calculation to obtain force!2. Stop3. Molecular dynamics step

]},[{min21),(

}{

2Ii

N

IIIII RERMRRL

i

rr&

rr& Φ−=

Φ∑

]},[{min)(}{ III RERU

I

rrΦ=

Φ

13

Car-Parrinello MD simulations *Car and Parrinello (1985) developed new method to...

• ”compute ground-state electronic properties of large and/or disordered systems at the level of state-of-the-art electronic structure calculations;

• perform AIMD simulations where the only assumptions are the validity of classical mechanics to describe ionic motion and the Born–Oppenheimer (BO) approximationto separate nuclear and electronic coordinates.“

]},[{21 2

Iii

ii

N

III RRML

r&&

r& Φ−ΦΦ+= ∑∑ εμLagrangian:

New: fictitous mass of electrons μ and constraint that orbitals stay orthogonal

( )∑ −ΦΦΛ+=Φij

ijjiijKS

Ii ER δε ]},[{r

14

CPMD equation of motion: *Nuclei:

Electrons:I

KS

II RERM r

r&&

∂∂

−=

∑ ΦΛ+Φ

−=Φj

jiji

KS

iE

δδμ &&

A two-component quantum-classical problem is mapped onto a two-component purely classical problem with employing the constraints that quantum mechanics

has to be fulfilled at all times.

Useful Literature:- Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Marx and Hutter, Cambridge University Press (2009)-Curr. Op. Chem. Biol. 2007, 11, 134; - Phys. Rep. 2007, 1-3, 1; -Top. Curr. Chem. 2007, 268, 133

What can be done:• 64-256 water (1 ns) usually: 20-50 ps• large computer resources• explore reaction and spontaneous events

15

Problems with theoretical methods

Ab initio or Car-Parrinello

simulations (CPMD)

Quantum Chemistry (QC)Molecular dynamics MD• Force field• transferability• pairwise additivity

• isolated molecules• expensive• 0~K, no environment

• expensive• simple QM methods

+

++

Useful Literature: Chem. Soc. Rev., 28 121-133, (1999); Curr. Op. Chem. Biol. 2007, 11, 134; Phys. Rep. 2007, 1-3, 1; Top. Curr. Chem. 2007, 268, 133

16

Outline

Introduction


AIMD simulations

MD simulations

Summary

17

Intermolecular forces in ILs (QC)

Stefan Zahn

Sebastian Lehmann

MatthiasSchöppke

MartinRoatsch

HenryWeber

Which intermolecular forces? Which shape and which form of potential? Which QC method? Are hydrogen bonds important?Relevant publications:Intermolecular forces: Angew. Chem. Int. Ed. 2008, 47, 3639; PCCP, 2008, 10, 6909Performance of DFT: JPCA, 2008, 112, 8430; E. I. Izgorodina, et al. JPCA, 2009, 113, 7064Hydrogen bonds: PCCP, 2010, 12 7473; J. Mol. Struct., 2010, 972 22Larger clusters: JCP, 2006, 124 174506

18

Comparison of IL and “normal” saltElectrostaticExchangeInductionDispersion

Results:

1. NaCl: No Dispersion

2. Dispersion for IL as large as induction!

Dispersion-correctedfunctionals work:JPCA, 2008, 112, 8430

NaCl, [Mmim][Cl]: on-top, in-plane

Angew. Chem. Int. Ed. 2008, 47, 3639

Energy decomposition using symmetry adapted perturbation theory SAPT(CCSD/aug-cc-pVDZ)

19

Total potentials and dimer potentials

• IL: Curves shallow• NaCl: Curve is steeper• IL: Electrostatics in the repulsive region!

TZVPP Basis

DispersionHFSCS-MP2

Blue: ILRed: NaCl

Angew. Chem. Int. Ed. 2008, 47, 3639 PCCP, 2008, 10, 6909

20

Hypothetic potentials

Angew. Chem. Int. Ed. 2008, 47, 3639 PCCP, 2008, 10, 6909

21

Performance of DFT

HF, MP2, B3LYP, BLYP-D, BP(DCACP)

Structure and energetics of the uncorrected functionalsare different to MP2

Dispersion-correctedfunctionals work:JPCA, 2008, 112, 8430see alsoE. I. Izgorodina, et al. JPCA2009, 113, 7064

22

27.625.116.426.235.7MAD-325.5-328.1-338.3-327.2-317.4ILV

-329.8-332.0-340.8-331.0-321.1ILI

B3LYPTPSSPBEBP86HF

-1.64.44.13.0MAD-358.8-357.0-355.8-360.5-361.9ILV

-348.1-350.5-353.2-354.7-353.3ILI

MP2BLYP-DBP86(DCACP)

BP86-DB3LYP-D

Performance of DFT

Useful Literature:IL and DFT: JPCA, 2008, 112, 8430; E. I. Izgorodina, et al. JPCA2009, 113, 7064

DFT-D: GrimmeJCC 2006, 27, 1787

DFT(DCACP):Lilienfeld et al. PRL 2004, 93, 153004.

23

Pauling 1960: “Under certain conditions a hydrogen atom is attracted by rather strong forces to two atoms, instead of only one, so that it may be considered to be acting as a bond between them.” and “it is now recognized that the hydrogen atom H, with only one stable orbital (the 1s orbital), can form only one covalent bond, that the hydrogen bond is largely ionic in character, and that it is formed only between the most electronegative atoms”

Pimentel/McClellan 1997: “A hydrogen bond exists when (1) there is evidence of a bond, and (2) there is evidence that this bond stericallyinvolves a hydrogen atom already bonded to another atom.”

Steiner/Sänger 1993: “…any cohesive interaction X-H…Y where H carries a positive and Y a negative (partial or full) charge and the charge on X is more negative than the one on H”

Hbond going back: What is known?

J. Mol. Struct., 2010, 972, 2224

Weak hydrogen bonds 1999: Lack directionality! But blue-shifted. X gains negative charge

Gilli Leitmotifs 1994: (±) CAHB double charge-assisted Hbonds: R-D-H…A-R‘ ↔ R-D- …H-A+-R‘

Arunan 2009: „The hydrogen bond is an attractive interaction between the hydrogen from a group X–H and an atom or a group of atoms Y, in the same or different molecule(s), where there is evidence of bond formation’. The most important criteria for a hydrogen bond are: (i) the H in the X–H group is more electropositive than X and (ii) the physical forces involved in hydrogen bonding should include attractive electrostatic forces, i.e. it should not be primarily dispersive forces.” CURRENT SCIENCE, 2007, 218; PCCP, 2009, 11, 8974

Hbond going back: What is known?

J. Mol. Struct., 2010, 972, 22

25

Hydrogen bonds can be present:- AlCl3-melts: Larger Anions (acidic mixtures Inorg. Chem., 2007, 47, 2751) → weaker

hbonds? → lower viscosities Wasserscheid and Keim, Aciee 2000, 39, 3777

- Close contacts between counter ions Seddon et al. Struct. Chem. 1990,1, 391

Hydrogen bonds are not important:- Lack of directionality Tsuzuki et al.PCCP, 2007, 9, 4780

- Time scale of transport properties are different than hbond dynamics

Hydrogen bonds are irritating:- Elimination of hydrogen bonds leads to increase in viscosity! PCCP, 2008, 10, 6909

→ We are dealing with a large confusion ofwhat hydrogen bond in ILs is

and whether it is important or not! J. Mol. Struct., 2010, 972 22 ; PCCP, 2010, 12, 7473

Hbond in ILs: What is assumed? HBond?

no HBond?

26

0.49148288.1-361.7[Emim][BF4]

0.77180264.8-33.0FH…H2O

160147141

a(XHY)

4.79300.6-400.5[Emim][Cl]2.14259.6-444.4[EtNH3][BF4]1.89258.3-470.5[NH4][BF4]

237.5-545.1[Na][Cl]

Δr(XH)r(XY)ΔEIP

ΔE in kJ/mol; r in pm; a in °, MP2/TZVPP

Y: acceptor atom

X-H…Y

X: donor atom

Favorite Donor!!!

Results:- ΔE shows: the larger

the ions are, the weaker IPs interact

- r(XY): > 280 pm for Emim → moderate

hbond ? - Donor-proton bond

elongated in all cases!- 180° angles not

found!

From typical salts to ILs: Geometry

PCCP, 2010, 12, 7473

27

Directionality in Ohno-type ILs and PILs

28

Charges on ion pairs

q in a.u.NH4

+ q(X): -0.818 q(H): 0.455EtNH3

+ q(X): -0.652 q(H): 0.434Emim+ q(X): 0.356 q(H): 0.237 HF: q(X): -0.559 q(H): 0.559

-0.154-0.7440.3070.204[Emim][Leu]

-0.020-0.9510.586-0.606H2O…FH

-0.023-0.6230.2850.393[Emim][BF4]

-0.002-0.5680.1130.988F3CH…FH

-0.137-0.056-0.048NCT

-0.8660.3130.333[Emim][Cl]-0.6180.478-0.698[EtNH3][BF4]-0.952-0.952[Na][Cl]q(Y)q(H)q(X)IP

Charge on - …acceptor mainly negativ, -…proton positive- …donor mainly more negativ than on isolated cation

- Exception is [Emim][BF4]- NCT = net charge transfer: Huge for [Emim][Cl]

PCCP, 2010, 12, 7473

29

Orbital pictureH2O…HF [Emim][Cl]

→ Orbital mixing→ HOMO decrease, LUMO increase (Emim][Cl])

→ Loss of electron density on XH bond!

PCCP, 2010, 12, 7473

30

HOMO-1 HOMO-2

HOMO-1: mainly Cl- anionHOMO-2: σ-type symmetry and interaction

between formally empty C2-H2 σ*-bond orbital and filled lone pair orbital of Cl-

Right: Difference between on-top and in-plane,Larger Anion: Decrease of CT and HBondIP → OT: CT unchanged; HBond vanishes

HOMO-1 and HOMO-2 of [Xmim][Cl]

Chem. Eur. J. 2010, 16, 8929PCCP, 2010, 12, 7473;

31

Summary to intermolecular forces• Intermolecular forces: Different forces are present and Ions act in the

repulsive if only electrostatics are considered dispersion forces are important!

• Density functional theory combined with correction schemes perform well!

• Hbonds can be present at the in-plane conformation in IM-IL and PILS!

• How likely are in-plane and the on-top conformers in the bulk liquid?

• If both conformers are populated, how often and how fast do these changes take place?

• What is the impact of possible hydrogen bonding? Does it favor the liquid or the solid state?

32

Outline

Introduction


AIMD simulations

MD simulations

Summary

33

Structure and fast (Hbond) dynamics of ILs(AIMD)

Jens Thar Martin Brehm

What is the structure of ILs? Is there a fast dynamics in ILs?Relevant publications:Pure IM-ILs: JPCB, 2009, 113, 15129Mixtures: JCP, 2008, 129, 104505; Inorg. Chem., 2007, 47 2751Gas phase simulations: unpublished work of Frank and PatriciaPILS: JCP, 2010, 132, 124506

Stefan Zahn Frank

Uhlig

Patricia Reuther

34

Ab initio molecular dynamics simulations

IL on film:

Show movies!

35

Simulations of 32 [C2mim][SCN]BOMD cutoff 300 Ryd, PBE, NVT, timestep = 0.5 fs, preequilibratedwith MD, ~10 ps simulations

• [SCN]− distribution differs at acidic protonJPCB, 2009, 113, 15129

36

Hydrogen bond dynamics of [C2mim][SCN]

Dynamics:< 0.3 ps

H2 always faster !S slower than N(exception H5)

hthh

tc)()0(

)( =

Luzar and Chandler, Nature, 379, 55, (1996)

⎩⎨⎧

=01

)(thIf h(0) hbonded and h(t) hbonded

If h(t) NOT hbonded

hbond: r(C-H…A)= 1st min RDFa(C-H…A) ≥ 150

JPCB, 2009, 113, 15129

37

[C2mim] and [SCN] contact dynamics

Without directionality:

H2 slowestH4/H5-N fastest

Dynamics: 5 ~ 10 ps

JPCB, 2009, 113, 1512938

Dynamics of [C2mim][SCN]With

directionality:< 0.3 ps

Without directionality:

5 ~ 10 ps

H2 always faster

H2 slowestH4/H5-N fastest

JPCB, 2009, 113, 15129

39

On-top or in-plane ?

47.2% in-plane 52.8% on-top




60-120 ° in-plane otherwhise on-top

40

CPMD Simulations of 48 [CH3NH3][NO3]CPMD cutoff 70 Ryd, Roethlisberger BP86-Dispersion, NVE, timestep = 4 a.u., preequilibrated with MD, ~20 ps simulations, 400 K

Hydrogen bonds are present!JCP, 2010, 132, 124506

41

Autocorrelation functions CX of ion pair conformationsC: continuous; R: One break before event happens

Life time of conformations

JCP, 2010, 132, 124506

42Autocorrelation functions of the hydrogen bond association

Life time of hbondC: same cation; A: same anionH: same proton; O: same oxygen

JCP, 2010, 132, 124506

43

Summary to AIMD simulation[Emim][SCN]:

• distinct cation-anion structure of the liquid

• fast unexpected dynamics

• on-top and in-plane both occur (probably equal likely);

[CH3NH3][NO3]:

• first dispersions-corrected AIMD simulation of IL

• hydrogen bond fingerprint in structure

• configurations dynamics

44

Outline

Introduction


AIMD simulations

MD simulations

Summary

45

Dynamical Heterogeneity in [Bmim][Br] (MD)

Miriam Kohagen

Jens Thar

Martin Brehm

GregorBruns

How “good” are quantum chemically derived charges for simulations?

46

SEN (=shared electron number) by Ernest R. Davidson.

The occupation number NA of atom A in a molecule AB is defined as: NA = tr D PA

and the shared electron number is σAB = NA + NB – nAB.

qtot : ± 0.8919 e., positive charge ≡ on carbon and nitrogen atoms of the ring.

Three schemes for charges

NBO (NBO=natural bond orbital) by Frank Weinhold.

NBO analysis = optimally transforming a given wavefunction into localized form, corresponding to the “lone pair“ and the “bonds“ of the Lewis structure picture.

± 0.8488 e., positive charge on the hydrogen atoms of the imidazolium ring.

RESP with HF/6-31G* and scaling with 0.8

47

Computational DetailsClassical MD-Simulation with GROMACS

500 ion pairs, Time step: 0.001 ps, T: 373 K

Nose – Hoover chain thermostats

Parameters from Amber-AA forcefield with some modifications for

imidazolium-based ionic liquids (Liu et al., J. Phys. Chem. B, 2004, 108, 12978).

Definition of the charges for the first simulation: RESP with HF/6-31G*

and scaling with 0.8 (Müller-Plathe)

Tg = 223 K

1. Simulation with an atomic charge of 0.8 e (20 ns) with RESP charges

2. 50 snapshots from the trajectory; subsequent SEN- and NBO-analysis

3. Simulations with new atomic charges (2 ns)

48

3D probability distributions of anions

RESP (0.8 e) NBO (0.85 e) SEN (0.89 e)

- Depending on the charge parametrization very different results!-Distribution of anions according to charge distributions. E.g. ring in

SEN system highest charged and ring protons almost not charged.-HBONDS: In-plane and on-top are both present!

49

3D-SDF of anions and cations

RESP NBO SEN

-Next neighbour cations prefer “free” space, because energetical benefit via π−π stacking

- SEN is exception, because anion occupies this space already!

50

Average speed mapped onto SDF

RESP NBO SEN

Blue: slow; Red: fast-Anions slow in areas with strong intermolecular forces

-Change from hydrogen bond state to “pure Colombic”state compare QC calculations: PCCP, 2010, 12, 7473

51

Polar and unpolar domains

C20-C20

H10-Br

- We observe Microheterogeneity!-Charge-effect is almost not present

- Left: Charge effect is present→ Proof for polar and unpolar

domains52

Intermolecular forces: Basics for micro-heterogeneity

Mircroheterogeneity(MH) from MD: from ethyl-, butyl-, hexyl-, decyl-[Mim][PF6] Costa Gomes et al. Top Curr Chem(2009) 290: 161–183

Origin of MH: Different intermolecular forces have different impact in different domains: Dispersion forces!Angew. Chem. Int. Ed. 2008, 47, 3639 PCCP, 2008, 10, 6909

53

Diffusion (Einstein) and Density

1.0616

1.0976

1.0530

Density

1.65

2.70

1.42

η[kg/(ms)]

0.14

0.03

0.26

[Br]-

anion

0.89

0.85

0.8

Charge

0.14SEN

0.03NBO

0.28RESP

[Bmim]+

cation

10-5

cm2 / s

- RESP fastest according to absolute charges- Cation faster than anion

- Numbers are in reasonable range compared to exp.

54

Comparison to experiment

- Temperature dependency of RESP

simulations-Diffusion is in excellent

agreement of measurements from

Naumov / Iacob / Kärger(pulsed field gradient

PFG NMR) and Sangoro/ Kremer (broadband dielectric spec. BDS)

- Synthesis and purification by

Lingenscheid / Giernoth

55

- With changing time scale, we observe different motions

-Side chain is faster than ring motion !

0.070.09

hbondcontinous

2.839.2211.527.73733.449.4287.932.1360

BuN-NIon pairhbondT[K] / τ[ps]

Dynamicalheterogeneity

56

Summary to [Bmim][Br] MD simulation• Results very dependent on force field parametrization → Still new

force fields are needed.

• Structure: Microheterogeneity is observed, distinction of polar and unpolar domains was proved ← Basics: Different intermolecular forces as shown by the QC calculations

• Dynamics: Excellent agreement with experiment!

• Dynamics: Heterogeneity found as well → Another origin for the liquid state (next to large and unsymmetric ions)?

57

Outline

Introduction


Calculations: MD-Simulations

Summary

58

Summary• Methods: - Severe approximations are made, -

lack of dispersion description can be corrected in DFT

• Intermolecular forces: - Different forces are present - and ions act in the repulsive if only electrostatics are considered weaker forces; Hbond is possible; - Dispersion!

• [Emim][SCN]: - distinct cation-anion structure of the liquid; - fast unexpected dynamics; - on-top and in-plane both occur (probably equal likely);

• [CH3NH3][NO3]: - hydrogen bond fingerprint in structure; -configurations dynamics

• [Bmim][Br]: Parametrisation important!; -Structural and dynamical microheterogenity;

• Correlation between MD and QC!

90° 0° -90°

Ene

rgy

in a

rbitr

ary

units

0

Angle between anion and imidazolium ring plane

[R1R2R3im][A][C1C1C1im][Cl][C1C1H1im][Cl][C2C1H1im][Cl][C2C1H1im][NTf2]

ionic liquids: from static quantum chemical calculations ... · frank jensen wiley & sons 2nd...

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