Stuff That You Think You Already Know but Actually ...

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In 1967, Uzi Kaldor published a paper that challenged the canonical treatment of multiple chemical bonds. ▷ He reasoned that maybe the “sigma pi” picture is ...

More Stuff That You Thought You Already Knew but Actually Probably Didn’t (As a Graduate Student)

(1.) Cory Camasta (Please excuse my use of the royal ‘we’)

[

It’s Periodic!

2𝑚

𝛻 2 + 𝑉 𝜓 ]𝜓 𝑞 = 𝐸𝜓 𝑞 = 𝑖ℏ Θ 𝑡 =𝑒

Electrons don’t complete a full orbit in 2π radians



−ℏ 2

−𝑖𝜓𝑡



Ψ 𝑞, 𝑡 = 𝜓 𝑞 Θ 𝑡



It takes two figurative cycles around the unit circle for an electron to return to its initial state



This equates to a period of 4π and is also the only math in this project that I can do all by myself



Think of a Mobius strip and this will make sense: In 2π radians, the electron returns the its initial position, however it is flipped “upside-down”. Another 2π radians returns the electron to its initial position and orientation.



The reason for this almost certainly has to do with the intrinsic spin of the electron (1/2)



The immediately-apparent relation of cycles per period as a function of spin (s) appears to be:

𝜕Ψ 𝜕𝑡

}

(2)

(The equations shown are just part of the derivation of the time-dependent Schrödinger equation and have little to do with the contents of this slide)5

0 to 2π 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑠 −1 𝑐𝑦𝑐𝑙𝑒𝑠 

But you should ask a particle physicist if you want more information ((this could be wrong))



Fun fact: s stands for sharp; p for principal; d for diffuse; and f for fundamental1

Levine, I. Quantum Chemistry, Third Edition. Allyn and Bacon, Inc. (1983) Weisstein, E. and Wolfram Research, Inc. (2015) 3 Wolfram Research, Inc. Mathematica, Version 10.0. Champaign, IL. (2014) 4 Susskind, L. Stanford University lecture, available online. (2014) 5 Adams, J. Personal interview. (2015)

(3)

1 2

0 to 4π

Quantum Tunneling 

This phenomenon may sound extremely complex but it is actually fairly simple.



A “quantum tunnel” is just a reaction path from reactant to product that deviates from the calculated minimum energy contour surface.



This occurs because a potential energy surface is defined as a function of only two internal coordinates.



An n-dimensional hypersurface is sufficient to account for a minimum energy path – good luck interpreting that, though.

Chemists have no excuses to forget the order of the planets (and Pluto is included!) Neptune

Neptunium

Uranium M e r c u r y

V e n u s

E a r t h

M a r s

J u p i t e r

S a t u r n

Plutonium Pluto Uranus

(Not to scale)

Chemical Bonding 

In 1967, Uzi Kaldor published a paper that challenged the canonical treatment of multiple chemical bonds



He reasoned that maybe the “sigma pi” picture is inaccurate and the atomic orbitals instead effectively combine to make a novel set of doubly-degenerate “banana bonds”

These are the canonical, nondegenerate molecular orbitals that represent the σ and π bonds:

MO 6 – σ

Optimized @ HF/cc-pVQZ, Isovalue 0.04

[HO]MO 8 – π

Chemical Bonding These are renderings of the degenerate “banana bonds” 1 formed by scaling the constituent orbitals by and then 2

taking their constructive and destructive combinations:

Destructive ( β- == σ – π )

Constructive ( β+ == σ + π )

*Note that subtracting σ from π instead of π from σ results in a bananabonding orbital with the same shape, but opposite phase. U. Kaldor, J. Chem. Phys., 46, 1981. (1967)

Chemical Bonding 

While all this was going on, the Greek train kept chugging down its tracks and lead to the acknowledgement of δ and φ bonds



The concept of non-canonical bonding has been somewhat accounted for with the m-centered, n-electron bond terminology which does not require any definite characterization of the “bond” that arises from the interaction



No matter which method you use to characterize chemical bonds, the probability density of finding an electron in any given place should not change and thus is the best physical representation that we have until we can view compounds at the atomic level with femtosecond (or finer) resolution

Ira N. Levine. Quantum Chemistry, Third Edition. Allyn and Bacon, Inc. (1983)

Electrostatic Potential: MP2(full)/VTZ-DHK(4) – 0.0004-0.04 e-/Å3, -0.05-0.05e (red→blue)

UwB97XD/VTZ-DHK(4) – 0.0004-0.04 e-/Å3, -0.05-0.05e (red→blue)

Spin Density: MP2(full)/VTZ-DHK(4) – 0.0004-0.04 e-/Å3

α

(Sources cited on final slide)

β

UwB97XD/DZVP2(5) – 0.0004-0.04 e-/Å3

α

β

Back to the bonds… HOMOs

E n e r g y

α

• The MOs explain everything that was shown on the previous slides (no NBO required!)

β

   

Five unpaired electrons β-spin on Cobalt α-spin on Chromium More electron density on Cobalt

• With this qualitative analysis, we can attempt to calculate the effective bond order… −



6 𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑒 + 3 𝑎𝑛𝑡𝑖𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑒 = − 3 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑒 3 𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑒 2 𝑐𝑒𝑛𝑡𝑒𝑟𝑠 Analysis based on occupied UwB97XD/VTZ-DHK MOs, Isovalue=0.04 e-/Å3



= Bond Order 1.5

Spinning in Circles (Observations) 

Spin contamination predicts more stable states in some calculations, gives garbage data in others



Isolated CrCoø prefers to exist in spin = 5/2 (sextet) state; nuclear distance ~2.5 Å



HF nuclear distance is the longest by ~0.4 Å; MP2 is close to DFT but longer by ~0.05 Å



Treating CrCo with DFT vs. MP2 theory changes results of ESP analysis and alignment of azimuthal spin



Stabilized HF spin density appears as a mix of MP2 and DFT densities, closer to MP2



Using triple-ζ basis vs. double- ζ results in very slightly more polarized spin density with improved Š2 values before and after projection



UwB97XD appears to describe the system more accurately than MP2



Together the metals contribute 15 electrons to a reportedly catalytic valence shell



Azimuthal spin and positive ESP on Cobalt supports its role as the catalytic metal



The high-spin d orbitals make up the lowest-energy α and β valence orbitals; at least one is bonding



Calculated “effective” bond order of 1.5 based on MO analysis is consistent with conclusions drawn in (1); they also reason a formal double bond with effective bond order 1.58 (using PBE method)



Effective bond order for optimized CrMnø complex is also consistent with (1), suggesting a formal quintuple bond (10 bonding electrons) minus 2 anti-bonding electrons that reduce the order to ~4.0, compared to 3.94 in (1); it is likely a sextet with nuclear distance ~2.1 Å, but greater spin contamination presents notable error – azimuthal spin moves to chromium in this isolated bimetallic



Spin-projected and stabilized Ψ often contain more spin contamination than raw SCF result – I cannot meaningfully interpret this yet



Quantum Chemistry of metal systems is still a clusterf*ck (even in 2015) 1

Clouston LJ, Siedschlag RB, Rudd PA, Planas N, Hu S, Miller AD, Gagliardi L, Lu CC. Systematic variation of metal-metal bond order in metal-chromium complexes. J Am Chem Soc. 135(35). (2013)

There is no Bromine in Theobromine (And just one more methyl group in Caffeine)

Knowledge drop: make damn sure that your “model” is accurate before pretending to know what you are talking about. Data for this “model” is given at right. Theobromine + CH4 → Caffeine + H2 Also, the use of Gcorr may be slightly misleading because, in this case, more negative means less stable.

~70 kJ/mol

**Use of SAMe and an N-methyltransferase as well as accounting for solvation will significantly reduce the free energy difference – maybe it will even become exergonic.

@ Method/cc-pVDZ (kJ/mol) Method

HF MP2(full)

ERxn 108.94 77.31

G°Rxn GcorrRxn 98.98 68.61

-9.96 -8.70

Based Chemistry 

Caffeine is synthesized from two different xanthine analogues: theobromine and paraxanthine



Methylation of paraxanthine and theobromine by SAMe (S-adenosylmethionine) is carried out by the eloquently-named enzyme Dimethylxanthine methyltransferase (2.1.1.160)3



The enzyme can also methylate 7-methylxanthine to theobromine



Theobromine is thought to be the major precursor to caffeine despite the fact that the enzyme has a higher affinity for paraxanthine3



It is a plantae alkaloid that activates pathogenesis-related genes, specifically the genes encoding pathogenesis-related protein 1a (PR-1a) and proteinase inhibitor II (PI-II), conferring resistance to various parasites2



May also disable cyclic Adenosine MonoPhosphate (cAMP) signaling in Manduca sexta (Tobacco horn worm) and other potential pathogens by inhibiting a phosphodiesterase1



It is found primarily in Coffea arabica (coffee), Cola nitida (cola), Ilex paraguaransis (maté) and Camellia sinensis (tea)1



Naming conventions in biochemistry are actually intuitive sometimes

Mikihiro Ogawa, Yuka Herai, Nozomu Koizumi, Tomonobu Kusano, and Hiroshi Sano‡. 7-methylxanthine Methyltransferase of Coffee Plants. J. Biol. Chem. 276 (11). (2000) 2 Yun-Soo Kim, Hiroshi Sano*. Pathogen resistance of transgenic tobacco plants producing caffeine. Pytochemistry 69. (2007) 3 SIB Swiss Institute of Bioinformatics. ExPASy: SIB bioinformatics resource portal. www.expasy.org (2015) 1

More B(3)ased Chemistry

Niacinamide Adenine Dinucleotide (+)

Niacin (VitaminB3)

Energies in kJ/mol

Method

Erxn

9.96 HF/6-31+G** HF//MP2(full)/ 5.90 Aug-cc-pVDZ MP2(full)/ Aug-cc-pVDZ -6.19

G°Rxn GcorrRxn 7.22

-2.74

3.63

-2.27

-8.87

-2.69

Niacinamide • We notice that ΔE and ΔG° are beginning to converge, however a higher level calculation should be completed before certain conclusions can be drawn • Given this data, we can still conclude that the reaction will be exergonic with an enzyme catalyst

Chemical Equilibrium

Fischer-consistent open-chain D-Glucose

Open-chain Glucose ΔE ~ 15.8 kJ/mol ΔG ~ 17.0 kJ/mol Keq ~ 0.60 ΔE ~ -26.2 kJ/mol ΔG ~ -13.4 kJ/mol Keq ~ 172.0 (This derivation of Keq may not be formally accurate1 but produces “better” numbers than the derivation using the sum of Gcorr and the total electronic energy, ε0 )

β-D-Glucose

ΔE ~ -33.7 kJ/mol ΔG ~ -20.0 kJ/mol Keq ~ 253.6

1.0 Kcal = 4.184 kJ Arrows read from left to right, Keq = 𝑒 −𝑅𝑇/∆𝐺𝑐𝑜𝑟𝑟 Energies and Keq reported for SMD(B3LYP/cc-pVDZ) optimizations, pictures at MP2(full) 1

Deakyne, C. Personal interview. (2015)

α-D-Glucose

More on Simple Sugars 

Glucose (a.k.a. dextrose) is the monomeric unit that was chosen to support life, while ribose is the monomeric unit that was chosen to encode life



The choice of ribose as the encoder makes sense because more information can fit in a smaller space this way – such is the natural progression of computing



This observation fosters the prediction that maybe the genetic code will evolve to incorporate more bases in the next few million years or so



α-Glucopyranose is the most stable conformer of glucose and also the most common conformer found in living tissue (glycogen and starch)



β-Glucose is the main component of cellulose, making up plant cell walls – it is more rigid than tissue composed primarily of α-glucose and thus cannot be digested by humans – what we call dietary fiber is mostly cellulose



In higher-order structures made from glucose monomers (i.e. glycogen, starch and cellulose), bonds between units are most commonly from the C4 –O(H) to C1. Branches in glycogen and starch are formed when the C6 –O(H) bonds to C1. The chains of these α-linked structures are helical with six monomers per turn.



Similar to glucose, ribopyranose, the six-membered ring form, is the most stable conformer of ribose; but β-ribofuranose, the five-membered ring form, is the form incorporated into RNA as is and DNA after dehydroxylation at the 2-carbon



Unlike glucose, the β-form of ribose is preferred in both furanose and pyranose forms of the sugar – this is likely due to the mass distribution



Fun fact: D-sugars and L-amino acids are the biologically-relevant forms

β-D-Ribofuranose

α-D-Ribofuranose Optimized @ MP2(full)/cc-pVDZ

Other Acknowledgements 1.

Gaussian 09, Revision D.01, Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian, Inc., Wallingford CT. (2009)

2.

GaussView, Version 5.0.9, Dennington, Roy; Keith, Todd; Millam, John. Semichem Inc., Shawnee Mission, KS, 2009.

3.

Head-Gordon, M; Pople, J. A.; and Frisch, M. J.; MP2 energy evaluation by direct methods. Chem. Phys. Lett., 153 503-06. (1988)

4.

F. E. Jorge, A. Canal Neto, G. G. Camiletti, and S. F. Machado, Contracted Gaussian basis sets for Douglas-Kroll-Hess calculations: Estimating scalar relativistic effects of some atomic and molecular properties, Journal of Chemical Physics 130, 064108. (2009)

5.

N. Godbout, D. R. Salahub, J. Andzelm, and E. Wimmer, Can. J. Chem. 70, 560. (1992)