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Tech

Parameter, Configuration, Argument

  • Parameter: to a function

  • Configuration: to an application

  • Argument: to a CLI command

Science

Polygon Rule, Huckel's Rule, bonding and antibonding

molecular orbitals (MOs), aromaticity

How to utilize the Polygon Rule and Huckel's Rule in practice:

For a 6 annulene:
By Polygon Rule: it has floorToOddNumber(6/2) => [3] bonding MOs.
By Huckel's Rule: 6 = 4*[1] + 2 => suggests AROMATIC
([3] - 1)/2 = [1]

※ remember that annulene is only applicable to the annulenes.

How to interpret aromaticity in terms of MO Theory:

Consider cyclohexane (C6H12):

Originally, each carbon atom has 2 p e-'s - that's 6*2 = 12 pi e-'s in total.
According to the Polygon Rule: Its "energy diagram" should be a hexagon, opening 6*2 spots for e-'s to settle in.
∵12 pi e-'s = 12 spots in the polygon ∴The polygon should appear "all occupied", including all the bonding and anti-bonding "MOs".
We noticed that the cyclohexane has no pi bond. (Does "anti-bonding" mean "canceling the bonding effect of the bonding MOs"?)

Now consider the [6]annulene (a.k.a. "benzene"):

One of the requirements for Huckel's Rule suggests that the molecule should have a continuous cyclic repetition of pattern "C-C=C", which actually reduces 1 e- from each C.
So, the total amount of pi e-'s in benzene = 6*2 - 6*1 = 6
It's just enough to fill up all the 3 bonding MOs.
(We observed 3 pi bonds in the benzene molecule. Does this suggests a "bonding MO" represents a pi bond in the molecule?)

Now consider cyclohexa-1,3-diene (benzene with one C=C bond reduced by H atoms):

2 additional pi e-'s compared to benzene
[Polygon Rule] the 2 additional e-'s are placed separately on the 2 degenerate LUMOs in the benzene diagram. Imagine these 2 e-'s are placed together in one of these LUMOs - an anti-aromatic MO is filled, so the molecule structure should have one pi bond less than benzene - and it truly is so.
[Huckel's Rule] 8 = 2*4 ===suggests===> anti-aromacitiy <===because that=== we are having 2 MOs with only 1 e- in each (can be proven by Polygon Rule), which makes up a "di-radical" structure.

Conclusion

The bonding/anti-bonding of the MOs - gives the amount of pi bonds in the molecule (just atom-and-atom bonds; not "pi conjugated system"):
a filled bonding MO suggest one more pi bond a filled anti-bonding MO suggest one less pi bond The Aromaticity - all about the stability and reactive electrons:
anti-aromaticity: has lone e-'s. They are likely to escape/grab e-'s from other molecules (a.k.a. "reactive"), thus making it very hard for this molecule to stay itself (a.k.a. "unstable").
aromaticity: has no lone e-'s. Every pi e- are paired in the MOs ,leaving no e-'s wandering about and causing troubles. It's not likely that this molecule change its structure, thus it has a high stability.

Resonance and Inductive Effects

There are two main electronic effects that substituents can exert:

  • RESONANCE effects are those that occur through the p system and can be represented by resonance structures. These can be either electron donating (e.g. -OMe) where p electrons are pushed toward the arene or electron withdrawing (e.g. -C=O) where p electrons are drawn away from the arene.

  • INDUCTIVE effects are those that occur through the s system due to electronegativity type effects. These too can be either electron donating (e.g. -Me) where s electrons are pushed toward the arene or electron withdrawing (e.g. -CF3, +NR3) where s electrons are drawn away from the arene.

"Thermodynamic reaction control" v.s. "Kinetic reaction control"

COMMON PROPERTIES

decide the composition of the product mixture of a reaction when:

  • competing pathways lead to different products

  • the reaction conditions influence the selectivity

Thermodynamic reaction control

  • Favors the thermodynamic product - the one with the lower internal energy.

  • Advantage in competition: product is more stable (due to lower U).

  • favored in lower temperature (where the Ea barrier cannot be easily overcome).

  • goes the thermodynamically-controlled pathway

  • e.g.: 1,4-addition of HBr to Dienes

Kinetic reaction control

  • Favors the kinetic product - the one with the lower kinetic energy.

  • Advantage in competition: goes faster (due to lower Ea).

  • favored in high temperature (where the Ea barrier can be easily overcome).

  • goes the kinetically-controlled pathway

  • e.g.: 1,2-addition of HBr to Dienes

Eigenvectors v.s. eigenvalues

English
Mandarin Chinese
English

For rhyming, the "feature vector" is hereinafter referred to as "eigenvector".

The eigenvectors and eigenvalues are used to describe a linear transformation.

The eigenvector describes the direction in which the direction does not change after the linear transformation is applied. The eigenvalues describe how much the vector will be stretched/compressed (as a scaling factor) in the direction above (the one that does not change direction after the linear transformation).

in case

  • V and W are finite dimensional, and

  • There are selected bases in these spaces,

Then: all linear transformations from V to W can be represented as matrices.

Under certain conditions (such as a linear transformation whose matrix form is a real symmetric matrix), the eigenvectors and eigenvalues can fully represent a linear transformation.

Mandarin Chinese

为了押韵,以下将“特征向量”称作“本征矢”。

本征矢 和 本征值 是用来描述一个线性变换的。

本征矢 描述了:作用该线性变换后,方向不会发生改变的方向。 本征值 描述了:上述(作用该线性变换后,方向不会发生改变的那个)方向上,向量会被拉伸/压缩多少(可以当作缩放系数来看)。

如果

  • V 和 W 是有限维的,并且

  • 在这些空间中有选择好的基,

则:从 V 到 W 的所有线性变换可以被表示为矩阵。

一定条件下(如其矩阵形式为实对称矩阵的线性变换),本征矢 和 本征值 可以完全表述一个线性变换。

Languages

Advance v.s. Advancement

Advance

Advancement

often associated with the idea of increased development or improvement.

As in: 'Advances in technology now make the laptop more popular than the desktop computer'

​

more of a long term, developing progression forwards.

As in: advancement in yo...