Diatomic Bond Order Calculator
 This tool computes bond orders of diatomic species with up to 20 electrons, including number of bonding, antibonding, and shared electrons, without using Molecular Orbital Theory (MOT), Lewis structures, or electron configurations.
 To use the tool, complete and submit form. Be sure to type in a diatomic chemical formula in the two large fields provided. Enter one chemical symbol per field. If the formula corresponds to a charged specie, type in its charge in the small topright field; otherwise, leave this field empty.
 This tool is not a chemical validator so be sure that the chemical formula you enter is valid, otherwise the results can be meaningless. Just because the tool returns a bond order, it doesn't necessarily mean that the computed bond order corresponds to a real chemical specie.
 We developed this tool inspired in Dr. Arijit Das set of innovative and time economic formulae for chemical education (Das, 2013; 2017; 2018; Das, Sanjeev, & Jagannadham, 2014). Das methodologies are suitable for computerbased learning (CBL) activities or for developing computer programs for solving chemistry problems. Our Hydrocarbons Parser is an example of this. In general, chemistry students who know how to write computer programs for solving chemistry problems have an edge when taking quantitative courses like analytical chemistry, instrumental analysis, chemometrics, computational chemistry, and similar courses. They are also better prepared for multidisciplinary research and postdoctoral work than those who cannot code.
 July 02, 2020 Update: We have added an Appendix section, which features the Almost Binary Heuristic. This is a much simpler heuristic for computing bond orders of diatomic species with up to 20 electrons. PHP code is included for those interested in building their own calculator. The heuristic also reproduces the socalled bond order "phone number" trick.

Bond Order Theory
Bond order (BO) has been defined as the number of chemical bonds between two atoms (Wikipedia, 2018a). This definition, though incomplete, is good enough for most diatomic species with up to 20 electrons. For instance, for H_{2} (HH), O_{2} (O=O), and N_{2} (N≡N), BO is 1, 2, and 3, respectively. Half values are also possible.However, that definition of bond order doesn't hold across all chemical species. One can do better by describing BO as an index of shared electrons. IUPAC's Gold Book describes BO as the electron population in the region between two atoms of a specie at the expense of electron density in the immediate vicinity of the individual atomic centers (McNaught & Wilkinson, 1997). Several authors consider IUPAC's definition a blurry one, and one that is more towards population analyses than towards counting shared electrons (Outeiral, Vincent, Pendás, & Popelier, 2018).
A BO close or equal to zero means that a very weak bond or no chemical bond is formed, even if said specie does exist. For instance, BO is 0 for Be_{2}, NaF, He_{2}, and Ne_{2}, but the first two do exist. Be_{2} is a very weakly bound molecule (Leach, 2018), with a reported bond length close to 2.44 angstrom (Lesiuk, Przybytek, Balcerzak, Musial, & Moszynski, 2019). NaF is an ionic compound composed of ions held together by electrostatic forces rather than by chemical bonds. In other words, all inexistent chemical species have a bond order of zero, but not all chemical species with a bond order of zero are inexistent.
So far, no stable neutral molecules involving covalently bound helium or neon are known. However, the "molecule" of two helium atoms bound by van der Waals forces does exist, with a "bond" that is about 5,000 times weaker than the covalent bond in the hydrogen molecule (Chemistry LibreTexts, 2016). He and Ne are noble gases, which can form molecular ions in the gas phase. The simplest example is the helium hydride molecular ion (HeH^{+}) discovered in 1925 (Wikipedia, 2018b).
Bond order theory is important because helps predict trends of other chemical features. For instance, BO's are proportional to bond strength, bond energy, and thermal stability, but inversely proportional to bond length (distance) and reactivity. Hence, BO is not a property, but an index used by chemists to understand other chemical trends.
In general, defining a bond order as "the number of chemical bonds between two atoms" is misleading. This definition is commonly used in introductory chemistry courses because it can be easily grasped by junior chemistry students, thought it can harm them later on in their career, or confuse them, to say the least. Consider this: The BO of the CN^{}, CN, and CN^{+} species are 3, 2.5, and 2, respectively. In each case, however, the C and N atoms are connected by a triple bond, sharing 6 valence electrons.
Chemistry teachers can do better by lecturing on BO as a measure of the relative strength of a bond rather than saying that BO is a bond count measure. Accordingly, we can do better by saying that, for instance, a specie with a BO equal to 1.5 is one with a bond (order) stronger than a single bond and weaker than a double bond.
 BO Methods
BO's are frequently predicted from Molecular Orbital Theory (MOT), first developed by Friedrich Hund and Robert Mulliken in 1933 (Roberts, 1961; Mulliken, 1970; Berry, 2000). Since then, several methods for counting BO's have been proposed: Coulson's chargebond order matrix, Wiberg index, ab initio bond order index, Becke's weight function, etc. Mayer reviewed some of these (Mayer, 2006, 1983).Additional models have been documented. For instance, Jules & Lombardi (2003) proposed an experimental BO method by inverting the Guggenheimer formula. Lu & Chen (2013) reported a BO analysis based on the Laplacian of Electron Density in Fuzzy Overlap Space. Szczepanik & Mrozek (2013) proposed a method based on quadratic bond order decomposition. RobyGould Bond Indices (RGBIs) were recently proposed to provide total bond orders by separately defining ionic and covalent bond indices (Gould, Taylor, Wolff, Chandler, & Jayatilaka, 2008; Alhameedi, Karton, Jayatilaka, & Thomas, 2018).
Manz (2017) documented what is probably the most comprehensive BO method to date. His method is based on the electron and spin magnetization density distributions, which works across chemical groups and periods, and with nonmagnetic, collinear magnetic, and non collinear magnetic materials with localized or delocalized bonding electrons. His method is implemented in the free Chargemol program. The program computes Density Derived Electrostatic and Chemical (DDEC) net atomic charges and atomic multipoles for periodic and nonperiodic systems. Manz pointed out that instead of assigning BO's in countable increments (e.g., whole numbers, halfnumbers, onethirds, etc), these should be allowed to vary continuously over the set of nonnegative real numbers since a material's geometry and electron cloud may be continuously deformed. This depicts the BO concept as a continuous function.
Outeiral et al. (2018) interpreted the delocalization index derived from Quantum Chemical Topology (QCT) as a bond order index. This index is constructed from quantum mechanical observables and is related to many modern chemical bonding interpretations like the Interacting Quantum Atoms (IQA) formalism.
In general, different definitions and counting methods can give rise to different BO values for the same chemical specie.
 Bond Order Heuristics
The above methods are perhaps too advanced for chemistry students. When teaching bond order theory to undergraduate, and even graduate students, most teachers prefer instead simplified heuristics, the most popular being(1)
where na is the number of antibonding electrons and nb the number of bonding electrons as determined from MOT. Since na ≤ nb, BO is nonnegative. This method works fairly well for diatomic species having up to 20 electrons.
When we developed this tool, we did so inspired in the following alternative method proposed by Inorganic Chemistry Professor Arijit Das:
(2)
where the absolute value or modulus of the difference between k and n is divided by 2 so BO remains nonnegative.
In (2), n is the total number of electrons of a diatomic specie and k is a parameter, set by Das as listed in Table 1 (Das, 2013; 2017; 2018; Das, Sanjeev, & Jagannadham, 2014).
Table 1. k, n values for Diatomic Species having up to 20 electrons. k 0 0 0, 4 4 4 4 4, 8 8 8 8 8 8 8 8 8, 20 20 20 20 20 20 20 n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 So for the following ranges of n: [0  2], [2  6], [6  14], and [14  20], Das k's are 0, 4, 8, and 20, respectively.
For borderline cases, occuring at n = 2, 6, and 14, Das set the k's to two possible bounds, k_{lower} and k_{upper}, where k_{lower} ≤ k_{upper}. These are listed in Table 1, in the cells colored in pale gray. Either one can be used without affecting the results, as absolute differences are taken. For instance, the bond order of N_{2} (n = 14) is three using either k = k_{lower} = 8 or k = k_{upper} = 20.
 Why Das Method Works
When we developed this tool, we were intrigued with the success of (2). We also wanted to know from where the parameter k and borderline cases come from as this is unclear from the above references.Stating the total number of electrons as n = na + nb satisfied our curiosity and helped us understand why this heuristic works as expected.
Using na = n  nb in (1), we obtained
(3)
So equating (2) and (3) reveals that
(4)
where (4) represents the number of shared electrons. As far as we can see from (4), Das method is a reformulation of (1) and (3).
It is evident from (4) that k, 2na, and 2nb are somehow related. To address this point, we inspected their parity. Without loss of generality, we derived two algebraic expressions from (4), namely
 k  n = 2nb  n = n  2na
 k  n = (2nb  n) = (n  2na)
Solving for k, two solutions are possible: k_{1} = 2na and k_{2} = 2nb. Thus, k_{1} ≤ k_{2} since na ≤ nb. Therefore, n = (k_{1} + k_{2})/2.
Table 2 summarizes our results.
Table 2. Results from (1)  (4). k 0 0 0, 4 4 4 4 4, 8 8 8 8 8 8 8 8 8, 20 20 20 20 20 20 20 n=(k_{1}+k_{2})/2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 na 0 0 0 1 2 2 2 3 4 4 4 4 4 4 4 5 6 7 8 9 10 nb 0 1 2 2 2 3 4 4 4 5 6 7 8 9 10 10 10 10 10 10 10 k_{1} = 2na 0 0 0 2 4 4 4 6 8 8 8 8 8 8 8 10 12 14 16 18 20 k_{2} = 2nb 0 2 4 4 4 6 8 8 8 10 12 14 16 18 20 20 20 20 20 20 20 k  2na 0 0 0, 4 2 0 0 0, 4 2 0 0 0 0 0 0 0, 12 10 8 6 4 2 0 2nb  k 0 2 4, 0 0 0 2 4, 0 0 0 2 4 6 8 10 12, 0 0 0 0 0 0 0 k  n,
2nb  n,
n  2na0 1 2 1 0 1 2 1 0 1 2 3 4 5 6 5 4 3 2 1 0 BO 0 0.5 1 0.5 0 0.5 1 0.5 0 0.5 1 1.5 2 2.5 3 2.5 2 1.5 1 0.5 0 Table 2 is enlightening:
 Das k's for the borderline cases are indeed the k_{1} = 2na and k_{2} = 2nb for the corresponding n. See Table 2 cells colored in pale gray.
 We may compute a k_{1} and a k_{2} for any n, not just for borderline cases.

Comparing Das k's with 2na and 2nb, it is clear that
k  2na and 2nb  k are multiples of 2 so we can write:
k ≡ 2na(mod 2) or "k is congruent to 2na modulus 2"
k ≡ 2nb(mod 2) or "k is congruent to 2nb modulus 2"This is not surprising since k_{1} and k_{2} also have the same parity.
 When computing BO's, it does not matter if we define the corresponding Das k's in terms of k_{1} or k_{2}. Therefore, we can safely remove all borderline cases from the first row of Tables 1 and 2 by retaining any of the two corresponding values, k_{1} or k_{2}.
 It seems that Das included borderline cases to graphically depict a continuous function over the range of n values considered. He also used a set of k's that works as a handy mnemonic for BO calculations. Table 3 lists this set in terms of k_{1} and k_{2}. We can modify this set by arbitrarily swapping k_{1} for k_{2} for the corresponding n, but the computed BO remains the same.
Table 3. Das k's listed as k_{1} and k_{2} values. k k_{1}=k_{2} k_{1} k_{1},k_{2} k_{2} k_{1}=k_{2} k_{1} k_{1},k_{2} k_{2} k_{1}=k_{2} k_{1} k_{1} k_{1} k_{1} k_{1} k_{1},k_{2} k_{2} k_{2} k_{2} k_{2} k_{2} k_{1}=k_{2} n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  New Formulae Derivable from Das Method
In this section we document some formulae derivable from Das heuristic. Using Das k's and (4) we can write(5)
(6)
(7)
(8)
(9)
and the number of antibonding and bonding electrons, as their electron ratios, cross products, and number of shared electrons ns can be easily computed. Furthermore, the number of unshared electrons nu is
(10)
which can be derived from nu = n  ns = (na + nb)  (nb  na) = 2na = k_{1} so the chemical meanings of k_{1} and k_{2} are evident. These formulae represent new applications for Das method.
 Tool Limitations
So far we have found one situation where (1) and (2), and therefore our tool, fails to produce a BO experimentally reasonable. This is the case of the monoflouride molecule (BF) and its ions. BF is isoelectronic with CO and N_{2} so (1) and (2) return BO = 3.However, experiments show that the bond order of BF is substantially less than 3. A value of 1.4 has been reported (Martinie, Bultema, Vander Wal, Burkhart, Vander Griend, DeKock, 2011), indicating a bond order between 1 and 2. It is true that three resonance structures can be proposed for BF, having respectively a triple, double, and single bond. However, its triplebonded resonance structure is not formally correct because requires of formal charges greater than ±1, in violation of the Pauling electroneutrality principle (Pauling, 1960). Martinie et al. also report BO = 2.6 for the CO molecule, while (1), (3), and other methods return BO = 3.
BF is a unique case, with its dipole moment inverted and fluorine having a positive formal charge even though it is the more electronegative atom. According to Fantuzzi, Cardozo, & Nascimento (2015), this is explained by the 2sp orbitals of boron being reoriented and having a higher electron density. BF is indeed a very special case.
That boron species are quite special is well known. MOT, as (1) and (2), predicts a single bond for the diboron molecule. However, a relatively recent study disputes this (Rashid, van Lenthe, & Havenith, 2017).
Before ending this section, we would like to underscore that our tool is not a chemical validator so submitting a chemically invalid formula may produce meaningless results. The tool works fairly well with diatomic species consisting of up to 20 electrons. Although a method for computing bond orders of polyatomic species is already documented (Jain, 2016), it is based on rigorously writing electron configurations and counting unpaired electrons. In the next section, we show that said method can fail.
 Beware of Blindly Relying on Chemistry Heuristics
From the above discussion, we can learn an important lesson: Chemistry is an experimental science. Therefore, chemistry heuristics can give you reasonable results until they fail. This is particularly true when such heuristics are based on outdated chemistry textbook ideas.As a mode of illustration, consider the case of the SO_{3} molecule. Robotically applying a bond order heuristic to this molecule can incorrectly give you a bond order of 2, by means of blindly counting electrons (Jain, 2016) and expanding the octet with d orbitals. In SO_{3}, however, S actually forms one double bond and two dative single bonds with oxygen atoms, resulting in a Yshaped specie with a bond order of 1.33, by means of Lewis resonance structures (Wikipedia, 2019a).
With regard to the idea, taught to entire generations of chemistry students, that elements beyond the second period can expand their octet by utilizing available d orbitals, we can only mention this: Since the 90's, quantum chemists have shown this idea to be experimentally incorrect as it is energetically unfeasible to use dorbitals for extra bonds (Kalemos & Mavridis, 2011; Durrant, 2015; Cowley, 2015; Northumbria, 2015). Indeed, the possibility of extensive dorbital participation has been discredited more than a quarter century ago (Reed & Schleyer, 1990; Magnusson, 1990). As back then Cooper, Cunnigham, Gerratt, Karadakov, & Raimondi stated in a JACS article published by the ACS (Cooper et al., 1994):
"Indeed, models based on d^{2}sp^{3}, dsp^{2}, and dsp^{3} hybrid orbitals are still in widespread use among professional chemists and are described in many of the most widely used textbooks. It is tempting to speculate as to why such models continue to survive when there is so much theoretical evidence which does not support them."
Therefore, heuristics based on the notion of dorbitals hybridization are likely to be misleading.
But, outdated ideas do not stop there. Consider the case of nitrogen. It has been claimed for a long time, by oldschool chemistry teachers and textbooks, that nitrogen cannot form five bonds. The experimental evidence shows this idea to be incorrect, exemplified by the diazomethane molecule (Durrant, 2015; Gerratt, Cooper, Karadakov, & Raimondi, 1997; Cooper, Gerratt, & Raimondi, 1989; Kurzydlowski & ZaleskiEjgierd, 2016; Science Daily, 2016; Wikipedia, 2019b), and by the nitrogen pentafluoride molecule (Emsley, 1990; Wikipedia, 2019c). Chemistry is indeed an experimental science.
The great thing about Chemistry as an Experimental Science is that experiments can debunk myths, hearsays, or just misleading ideas. In this ongoing, always updated, piece of article, we have debunked few of them:
 Bond Order is the number of bonds between two atoms. Nope.
 A chemical specie cannot exist if its bond order is zero or close to zero. Wrong.
 The bond order of BF is 3. Wrong.
 The bond order of SO_{3} is 2. Wrong.
 Nitrogen cannot form five bonds. Wrong.
 Octet expansion is the result of utilizing available d orbitals. Nope. Wrong. Nope.
Of those, the most troubling is the last one since it appears to be the result of not knowing, ignoring, or dismissing experimental research results by some almostsciencecult sectors of the chemical community.
 Conclusion
Das method is a reformulation of (1), effectively working as a shortcut for computing bond orders. We identified the origins of Das k's and of the borderline cases as well as the fact that k, 2na, and 2nb are all congruent numbers.Using (2)  (4) we derived (5)  (10): a new set of formulae for predicting number of antibonding (na) and bonding (nb) electrons, including their ratios, cross products, shared and unshared electrons, and without any further references to MOT!
We identified k_{1} and k_{2} values for all diatomic species having up to 20 electrons, not just for the borderline cases at n = 2, 6, and 14. The chemical significance of k_{1} and k_{2} was also identified, where k_{1} = 2na = nu, k_{2} = 2nb, n = (k_{1} + k_{2})/2, and ns = (k_{2}  k_{1})/2; also, ns = n  k_{1}.
Working on this tool was really gratifying. It inspired us to derive a novel algorithm for predicting number of unpaired electrons and magnetic properties of single atoms, diatomic species, and their ions. Hopefully, this algorithm will be available soon in the form of a new chemistry calculator.
Last but not least, we are currently working on a tool that computes bond orders for all kind of chemical species with any number of electrons, without having to use MOT, Lewis structures, or electron configurations.
 Appendix  The Almost Binary Heuristic: Yet another heuristic that still fails
Written on July 02, 2020I deviced this mnemonic, the Almost Binary Heuristic, after inspecting the last row of Tables 1 and 2. Notice that no k values are used. It all boils down to writing a new table as given below.
Table 4. The Almost Binary Heuristic n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 p 0 1 0 1 0 1 0 1 0 1 0 2^{p} /+ 1 0 1 0 1 0 1 2 3 2 1 0 BO 0 0.5 1 0.5 0 0.5 1 0.5 0 0.5 1 1.5 2 2.5 3 2.5 2 1.5 1 0.5 0 In the first row of Table 4, we list the total number of electrons n of diatomic species, up to 20 electrons. In the second row and for n even, we associate a value p that alternates between 0 and 1 so we end up with an almost binary sequence of p values.
In the third row and for n even, we make the assertion that if n is equal to 12, 14, or 16, the bond order is given by 2^{p} + 1; otherwise by 2^{p}  1. To convince yourself of this, do the calculations.
Finally in the fourth row and for n odd, the bond order is the average of neighboring bond orders!
Thus for H_{2}^{+}, n = 1 and BO = 0.5; for NO, n = 15 and BO = 2.5; and so forth.
Some on the Web (Quora, 2017) call the BO sequence for n even the "phone number" trick, credited to Bhuvnesh Mittal (Mittal Sir) BMC Classes; however, they seem to start the sequence at BO = 1, n = 2.
We reproduced the "phone number" trick, but starting at BO = 0, n = 0, with the following php script.
<?php //Keep this and next line in place for proper credit given. //Copyright 2020, Dr. Edel Garcia, Minerazzi.com $p=1; for($i=0;$i<21;++$i){ if($i % 2 == 0){ $p=1$p; if($i==12$i==14$i==16){ $BO=pow(2, $p) + 1; } else{ $BO=pow(2, $p)  1; } echo "BO = ".$BO." → n = ". $i . "<br />"; } } ?>
The script output is given below.
BO = 0 → n = 0
BO = 1 → n = 2
BO = 0 → n = 4
BO = 1 → n = 6
BO = 0 → n = 8
BO = 1 → n = 10
BO = 2 → n = 12
BO = 3 → n = 14
BO = 2 → n = 16
BO = 1 → n = 18
BO = 0 → n = 20
Feel free to copy/rewrite the script with your favorite programming language or use it to build your own bond order calculator tool. Just please keep the credit lines in place. :) We leave as a homework the coding of bond orders for n odd.
As the Almost Binary Heuristic reproduces the bond orders obtained in Table 2, it fails when the heuristics given by (1) and (2) fail. RIP.
Dr. Edel Garcia
 Chemistry educators, scholars, and students interested in bond order theory and its applications.
 Calculate the terms listed in Table 2 for the following:
HeH^{+}, Be_{2}^{+}, Li_{2}^{2}, B_{2}^{}, CN^{}, O_{2}^{+}
 We are sincerely in debt to Dr. Arijit Das from Ramthakur College, Agartala, West Tripura, India for encouraging us to develop this tool for educators, scholars, and chemistry students.
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